Forex Trading and the WMR Fix
Martin D.D. Evans⇤
August 2014
First Draft
Abstract
Since 2013 regulators have been investigating the activities of some of the world’s largest banks around
the setting of daily benchmarks for forex prices. These benchmarks are a key linchpin of world financial
markets, providing standardize prices used to value global equity and bond portfolios, to hedge currency
exposure, and to write and execute derivatives’ contracts. The most important of these benchmarks,
called the “London 4pm Fix”, “the WMR Fix” or just the “Fix”, is published by the WM Company
and Reuters based on forex trading around 4:00 pm GMT. This paper undertakes a detailed empirical
analysis of the how forex rates behave around the Fix drawing on a decade of tick-by-tick data for 21
currency pairs. The analysis reveals that the behavior of spot rates in the minutes immediately before
and after 4:00 pm are quite unlike that observed at other times. Pre- and post-Fix changes in spot
rates are extraordinarily volatile and exhibit strong negative serial correlation, particularly on the last
trading day of each month. These statistical features appear pervasive, they are present across all 21
currency pairs throughout the decade. However, they are also inconsistent with the predictions of existing
microstructure models of competitive forex trading.
Keywords: Forex Trading, Order Flows, Forex Price Fixes, Microstructure Trading Models
JEL Codes: F3; F4; G1.
* Georgetown University, Department of Economics, Washington DC 20057 and NBER. Tel: (202) 687-1570 email:
evansm1@georgetown.edu.
1
Introduction
In the summer of 2013 the financial press reported the existence of numerous regulatory investigations into
the foreign currency (forex) trading activities of some of the world’s largest banks. These on-going investigations by the European Commission, Switzerland’s markets regulator Finma and the country’s competition
authority Weko, the UK’s Financial Services Authority, the Department of Justice in the US, the Hong Kong
Monetary Authority and the Australian Securities and Investment Commission, among others, center on the
actions of the banks’ forex traders around the time that benchmark currency prices are determined. The
most widely used benchmarks are provided by the WM Company and Reuters, based on forex transactions
around 4:00 pm GMT. These benchmarks are colloquially known as the “London 4pm Fix”, “the WMR
Fix” or just the “Fix”. In June 2013 Bloomberg News reported that some forex traders at the world’s
largest banks had been allegedly colluding in an attempt to manipulate the Fixes, and that regulators were
investigating the matter. Since then, very little information concerning the investigations has been made
public.1
Benchmark interest rates and forex prices, like LIBOR and the Fix, are key linchpins of the world’s
financial markets. In particular, the Fix provide standardize currency prices that are used to value global
equity and bond portfolios, to (dynamically) hedge currency exposure, to write and execute derivatives
contracts, and administer custodial agreements. In light of this, the fact that so many financial regulators
are investigating forex trading around the Fix suggests that the allegations of collusion are credible. What is
much less clear is whether collusion, if indeed it took place, could have materially a↵ected the determination
of the Fix to the detriment of participants in the forex and other financial markets. This paper presents
statistical evidence pertinent to this issue. In particular, I used a decade’s worth of tick-by-tick data from 21
currency paris to study the behavior of the forex prices around the Fix. To be clear, this analysis does not
provide any direct evidence on the allegations of the collusion being investigated by regulators. Instead it
documents a set of facts about the behavior of forex prices around the Fix which may be juxtaposed against
models of forex trading.
The sine qua non of the Fix is that it provides an accurate measure of the prices (i.e., spot rates) at
which currency pairs trade around a specified time (4:00 pm GMT)2 . This is true in the narrow sense that
each Fix is computed from transaction prices in a currency pair during a 60 second window around 4:00
pm. But, interpreted more broadly, it is not the case. The central finding of my analysis is that the Fix
benchmarks are very unrepresentative of the prices at which currency pairs trade in the hour or so around
4:00 pm. This finding holds true in all 21 currency pairs I examine (including the major currency pairs: e.g.
USD/EUR, CHF/USD, USD/GBP and JPY/USD), and for every year between 2004 and 2013. It is also
particular striking on the last trading day of every month. Initial news reports concerning the allegations
of collusive behavior of banks’ forex traders around the Fix showed instances where the prices from forex
trades immediately around 4:00 pm looked very di↵erent from the prices several minutes earlier or later.
My analysis shows that these examples of price movements around the Fix are far from unusual. On the
contrary, they have been commonplace throughout the span of my data.
1 There have been several news stories reporting the dismissal of forex traders from major banks, but the reasons behind
these dismissals - particularly with respect to the regulators’ investigations - were not disclosed.
2 Hereafter, all times refer to GMT.
2
My main findings are most easily summarized with the aid of a plot. Figure 1 shows the average paths
for the USD/GBP spot rate during the 15 minutes before and 30 minutes after the 4:00 pm.3 The solid
lines plot the average level of spot rates measured in basis points relative to their level at 3:45 pm from all
end-of-month trading days between the start of 2004 and end of 2013. The dashed lines depict the analogous
plots from all other (i.e. intra-month) trading days. The upper branch of the solid and dashed plots shows
the average spot rate level on those days when rates rose in the 15 minutes before the Fix, the lower branch
shows the level when rates fell.
Figure 1: USD/GBP Spot Rate Profiles Around the Fix
20
15
10
5
0
−5
−10
−15
−20
−15
0
15
30
Notes: Solid lines plot the average path for the USD/GBP from 15 minutes before to 30
minutes after the 4:00 pm GMT from all end-of-month trading days between the start
of 2003 and end 2013. The dashed lines plot the average path over the same interval
on all other (intra-month) trading days. Paths are plotted in basis points relative to the
USD/GBP rate at 3:45 pm GMT.
Several features of the plots in Figure 1 are representative of my main findings. The first concerns the
di↵erence between the level of the Fix and the prior level of spot rates. Figure 1 shows that relative to
the 3:45 level, this di↵erence is roughly ±15 basis points on average at the end-of-the month, and ±7 basis
points on intra-month days. I refer to these di↵erences as the pre-fix rate changes. My analysis shows
that rate changes of these magnitude are very rare in normal trading. I use the eleven year span of the
tick-by-tick data to construct precise estimates of the distribution of rate changes that arise in forex trading
away from significant (recurrent) events, such as the Fix and the scheduled release of macro data. These
estimated distributions summarize the behavior spot rates under “normal” trading conditions, and can be
3 Hereafter I use the term “spot rate” when referring to the price at which a particular currency pair trades. The USD/GBP
spot rates plotted here are computed from the mid-point of the bid and o↵er rates, see Section 2 for details.
3
used to calibrate the rate changes we observe in the minutes leading up the Fix. This calibration exercise
reveals that the pre-fix rate changes routinely seen at the end of each month fall in the extreme tails of
the rate-change distribution based on normal forex trading. For example, in the case of the USD/GBP, the
change in rates between 3:45 and 4:00 at the end of each month appear in 95th percentile of the rate-change
distribution six times more frequently than we see under normal trading conditions. This pattern applies
across all the currency pairs, and across horizons ranging from one hour to one minute before the Fix. It is
also evident, to a lesser degree, in the intra-month data. As Figure 1 shows, intra-month pre-fix rate changes
are on average smaller than their end-of-month counterparts, but they still appear in the 95th. percentile
of the rate-change distribution four times more frequently than in normal trading. In sum, the movements
in spot rates leading up to the 4:00 pm Fix are extraordinarily volatile across all time periods and currency
pairs.
My second main finding concerns the relation between spot rates leading up to 4:00 pm, the Fix benchmark, and rates after 4:00 pm. The plots in Figure 1 show that the average path for the USD/GBP spot
rate at the end of the month slope in opposite directions either side of (a point close to) the 4:00 pm Fix.
In other words there are partial reversals in rate changes around the Fix: on average rates tend to fall after
rising towards the Fix, and rise after falling towards the Fix. These reversals are larger in end-of-month
than intra month data (as shown in Figure 1) and are present in the rate-dynamics of all 21 currencies
studied. Like the pre-fix rate changes, unusually large post-fix changes (i.e., rate changes from the Fix going
forward) regularly occur at the end of each month. In the 15 minutes following the Fix they appear in the
95th percentile of the rate-change distribution at two to four times the rate we see under normal trading
conditions. Statistically, reversals show up as negative correlations between pre-fix and post-fix rate changes.
I find evidence of large statistically significant negative correlations for most currency pairs in end-of-month
data over horizons ranging from one to 15 minutes. These findings stand in sharp contrast to the very small
degree of serial correlation in the rate changes generated by normal forex trading.
The statistical evidence overwhelming indicates that for all currency pairs the behavior of spot rates
around the Fix is very unusual. These findings have several important implications. First, they undermine
the notion that the Fix benchmark provides a snapshot of the spot rates (forex prices) associated with
normal trading activity during the day. This notion is implicit in the widespread use of the Fix as the “daily
spot rate”. In reality, however, the daily range for spot rates is similar in size to the time series changes
in Fix benchmarks over months, quarters and longer. Moreover, Fix benchmarks generally fall towards the
extremes of the daily range for spot rates. Together, these findings imply that the forex returns computed
from the Fix benchmarks often materially di↵er from the returns on forex positions that were initiated
and/or closed at times away from 4:00 pm on the same days. This means that the returns routinely studied
in the international finance literature (computed from the Fix benchmarks) are at best noisy estimates of
the returns achieved by actual investors.
My statistical findings also present a challenge to theories of trading behavior around the Fix. As Section
1 explains, there are particular institutional factors that weigh on the trading decisions of market participants
around the Fix that are not present at other times during the trading day. These factors figure prominently
in the anecdotal accounts of forex trading around the Fix reported in the financial press, but it is unclear
whether such trading can account for the unusual behavior of spot rates we observe. Similarly, existing
4
microstructure models of the forex trading are silent on whether the unusual statistical characteristics of
spot rates around the Fix can arise in an equilibrium when these institutional factors are present.
Currency trading around the WMR Fix has not been the focus of academic research, with the notable
exception of Melvin and Prins (2011). They describe how currency hedging by portfolio managers generate
forex trading around the Fix. Their empirical analysis focuses on the links between forex and equity returns
in the G10 currencies between 1996 and 2009, particularly the e↵ects of equity returns on forex volatility
around the Fix. This paper provides a more detailed examination of the behavior of spot rates round the
Fix across a wider rage of currency pairs that compliments the analysis in Melvin and Prins (2011).
The remainder of the paper is structured as follows. Section 1 describes the institutional details of the
WMR Fix and discusses the implications of existing theoretical trading models for the behavior of spot
rates around the Fix. Section 2 describes the data. My empirical analysis begins in Sections 3 and 4. Here
I examine how the Fix benchmarks relate to the daily variations in spot rates, and document how rates
behave under normal trading conditions. Sections 5 and 6, in turn, examine the behavior of spot rates in
the minutes before and after 4:00 pm. Finally, in Section 7, I examine the trading implications of the spot
rate reversals around the Fix. This analysis places an economic perspective on my statistical findings, and
provides indirect evidence on the degree of competition in forex trading around the Fix. Section 8 concludes.
1
Background
1.1
Institutional Background
The WMR Fix was established as a key financial benchmark at the end of 1993. Morgan Stanley Capital
International (MSCI) announced that from December 31st 1993 onwards it would use the benchmark forex
prices compiled at 4:00 pm GMT by the WM Company and Reuters to value the foreign security positions
in its MSCI equity indices4 – indices widely used track the performance international equity portfolios.
Since then, the Fixes have been incorporated into numerous other tracking indices5 and derivatives6 . WRM
Fixes are the de facto standard for construction of indices comprising international securities. They are also
routinely used to compute the returns on portfolios that contain foreign currency denominated securities
as well as the value of foreign securities held in custodial accounts. WMR Fixes are now computed every
half-hour for 21 currency pairs and hourly for 160 currency pairs, but the 4:00 pm Fix remains the single
most important benchmark forex price each day. My analysis focuses exclusively this particular benchmark.
Although forex markets operate continuously, trading activity is heavily concentrated around European
business hours for most currency pairs (exceptions include Asian currencies where trading is concentrated
earlier in the day). Thus the 4:00 pm Fix occurs towards the end of the daily window where there are a large
number of potential counterparties available to participate in forex trades for major currency pairs. This is
an important feature of the Fix. Market participants wanting to trade in the minutes following the Fix will
4 Initially, the Fix benchmarks were used to compute the MSCI indices for all but the Latin American countries. After 2000
they were used for all the country indices.
5 Recent examples include: Dow Jones Islamic Market, Global Real Estate (FTSE EPRA/NAREIT) and Global Coal (NASDAQ OMX) indices.
6 See, for example, the USD volatility warrants issued by Goldman Sacks; Wiener Borse AG fInancial futures and CME spot,
forward and swaps.
5
face spreads between bid and o↵er rates o↵ered by potential counterparties that are comparable to spreads
earlier in the day, but in the next hour or so (with exact timing depending on the particular currency pair)
spreads widen as the number of counterparties shrinks. Generally speaking, forex trading becomes increasing
costly (in terms of spreads) as one moves later into the day past the 4:00 pm Fix.
The Fix is computed over a one minute window that starts 30 seconds before 4:00 pm. The methodology
is described on the WMR website (http://www.wmcompany.com) as follows:
Over a one-minute Fix period, bid and o↵er order rates from the order matching systems and
actual trades executed are captured every second from 30 seconds before to 30 seconds after the
time of the Fix. Trading occurs in milliseconds on the trading platforms and therefore not every
trade or order is captured, just a sample. Trades are identified as a bid or o↵er and a spread is
applied to calculate the opposite bid or o↵er.
Using valid rates over the Fix period, the median bid and o↵er are calculated independently
and then the mid rate is calculated from these median bid and o↵er rates, resulting in a mid
trade rate and a mid order rate. A spread is then applied to calculate a new trade rate bid and
o↵er and a new order rate bid and o↵er. Subject to a minimum number of valid trades being
captured over the Fix period, these new trade rates are used for the Fix; if there are insufficient
trade rates, the new order rates are used for the Fix.
Two aspects of this methodology are noteworthy. The first concerns the source of the bid and o↵er forex
rates. The electronic trading platforms run by Reuters and Electronic Broking Services (EBS) (now owned by
ICAP) are the main trading venues for dealer-banks in the forex market. EBS is the primary trading venue for
EUR/USD, USD/JPY, EUR/JPY, USD/CHF and EUR/CHF, and Reuters Matching is the primary trading
venue for commonwealth (AUD/USD, NZD/USD, USD/CAD) and emerging market currency pairs.7 The
WMR Fix uses either platform as the primary data source depending on the currency pair, and rates from
Currenex as a secondary source for validation. In recent years forex trading platforms have proliferated
so that a wider array of (tradable) bid and o↵er rates are available to market participants than just those
sourced by the Fix methodology. Thus the Fix should be viewed as a benchmark computed from a subset
rather than the universe of forex rates available in the one minute window around 4:00 pm.
The second aspect concerns the computation of the trade benchmark. A careful reading of the methodology reveals that no account is taken of trading volume. This means that the transaction price recorded as
the result of the submission of a market order to buy or sell forex valued at 20 million USD has exactly the
same weight in computing the benchmark as an order valued at 200 million USD. Moreover, the methodology
takes no account of order flow (i.e., the di↵erence between the value of market orders to buy forex and sell
forex within a time interval). Order flow during the one minute Fix window may be strongly positive or
negative, but this fact will not be reflected in the Fix benchmark (provided there are enough buy and sell
market orders to compute the median bid and o↵er trade rates).
The existence of the 4:00 pm Fix per se would not be of any great significance were it not for the fact
that market participants face strong economic incentives to trade forex in and around the Fix window. It
7 Throughout, I use market abbreviations for currencies: e.g., U.S. Dollar (USD), Euro (EUR), Swiss Franc (CHF), Japanese
Yen (JPY), British Pound (GBP), Australian Dollar (AUD), Canadian Dollar (CAD) and New Zealand Dollar (NZD). I also
follow market conventions when quoting spot rates in direct or indirect form, e.g. EUR/USD rather than USD/EUR.
6
is hard to overstate the importance of this point. If the Fix were calculated every day according to the
methodology described above and archived as a data series, its existence would have no economic relevance
for the behavior of the forex market. Fixes would simply be snap shot measures of forex rates around 4:00
pm that could be useful for research. One could argue about whether the methodology could be improved,
but these would be arguments about measurement rather than arguments about how the existence of the
Fix a↵ected actual market activity. Of course, in reality, the Fixes aren’t simply archived. Instead they are
used in real time to value other securities, such as equity portfolios and derivatives. Market participants face
strong incentives to trade in and around the Fix precisely because the Fixes are used for real-time valuation
purposes.
The trading incentives created by the existence of the Fix originate with two groups of market participants.
The first comprises investors wishing to hedge some of the currency risk associated with their holdings. As
Melvin and Prins (2011) stress, fund managers with cross-boarder equity investments are important members
of this group. Because the performance of their investments are often tracked against the returns on the
MSCI indices that use the Fix, many managers will want to reduce the tracking error of their own portfolios
by choosing to hedge some of their (forex) exposure to the Fix. In principle this hedging could take place
continuously through the adjustment of forex forward positions, but in practice most managers adjust their
currency hedge positions once a month, usually on the last trading day of the month. This hedging activity
produces orders to purchase or sell forex. And, since the managers are concerned with tracking the MSCI
indices, they want their forex orders to be filled at the Fix to minimize the tracking error in their own
portfolio’s performance.
As a concrete example, suppose the UK based mutual fund manager holds part of his portfolio in US
equities. At the end of last month the US position had a value of 1 billion USD. The manager also maintains
a 50 percent forex hedge ratio against this position, which was short 500 million USD at the end of last
month. Now suppose that the value of the US equity portfolio rises by five percent during the current month
to a value of 1050 million USD on the day prior to the end of the month. In this situation, the manager
would want to increase his short USD position by 25 million, so on the last day of the month he would
place an order to sell 25 million USD with a dealer-bank. This order could be submitted as a standard
forex order, to be filled immediately by the dealer-bank at the best bid rate for the USD/GBP prevailing in
the market (say on Reuters Matching). Alternatively, the manager could submit a “fill-at-fix” forex order,
which specifies that the order to sell 25 million USD should be filled at the Fix benchmark rate established
at 4:00 pm.8 By market convention, fill-at-fix orders must be submitted to dealer-banks before the 3:45
pm. Consequently, the submitter of such an order faces a good deal more uncertainty about the exact rate
at which the order will be filled than with a standard forex order.9 Nevertheless, a fill-at-fix order will be
preferable to the fund manager because it guarantees that the GBP value of the adjusted hedge portfolio
matches 50 percent of the equity position valued in GBP at the new USD/GBP Fix benchmark.
This example illustrates how the use of the Fix in valuing equity portfolios combines with the desire of
fund managers to (partially) hedge forex risk to produce fill-at-fix forex orders leading up to the Fix. The
8 The actual rate received by the manager will also include a spread adjustment to the Fix benchmark depending on whether
the order was to buy or sell foreign currency. The fill-at-fix contract may specify the spread reported by WMR or one set by
the dealer-bank.
9 As we shall see, the volatility of spot rates between 3:45 and 4:00 pm is several orders of magnitude higher than the volatility
of rates during the (fraction of) seconds between the submission and filling of a standard forex order.
7
use of the Fix benchmarks in derivative contracts produces a similar incentive to submit fill-at-fix orders
from other investors wishing to partially hedge their derivative positions. Thus, the existence of the Fixes
and their use in real-time valuation produces a hedging incentive for the submission of fill-at-fix orders before
3:45 pm. These incentives are particularly strong at the end of the month.
The second group of market participants a↵ected by the Fix are the dealer-banks that accept fill-at-fix
forex orders. As noted above, fill-at-fix orders di↵er from standard forex orders because the dealer-banks
agree to fill them at the Fix benchmark rate at least 15 minutes before that rate is determined. Thus,
in e↵ect, the dealer-banks are o↵ering a guarantee that the order will be filled at particular point in time
whatever the prevailing rates (as represented by the Fix) might be.10 By contrast, in accepting a standard
forex order the dealer-bank undertakes to fill the order immediately at the best available prevailing rate.11
Of course, such guarantees represent a source of risk to the dealer-bank. Generally speaking, it is the desire
to manage this risk that creates incentives for dealer-banks to trade in and around the Fix.
To understand these risk, consider the position of a dealer-bank that by 3:45 pm has on net fill-at-fix
orders to purchase 200 million GBP in the USD/GBP market. Broadly speaking, there are three strategies
available to the dealer-bank. The first is simply to fill the fill-at-fix orders immediately at the prevailing
market rate. This strategy runs the obvious risk that the Fix benchmark will be established at a significantly
di↵erent level than current rates. In this particular example, the dealer risks a fall in the USD/GBP rate
between 3:45 and 4:00 pm, which would produce a (USD) trading loss because the 200 million GBP purchased
at 3:45 would be sold on to the bank’s fill-at-fix customers at a lower USD price established by the Fix. The
second strategy is to purchase the 200 million GBP at a rate as close as possible to the Fix benchmark. This
involves trading within the one minute Fix window, and even then, there is no guarantee that the actual rate
at which the GBP purchase is made exactly matches the Fix benchmark (because the latter is an average
of rates during the Fix window). The third strategy has two prongs: (i) purchase the 200 million GBP
incrementally between 3:45 and 4:00 and (ii) take a speculative position in anticipation of a change in rates
between 3:45 and 4:00. The first prong reduces the risk from a fall in the USD/GBP rate relative to the
first strategy, but it cannot eliminate the risk entirely. Goal of the second prong is produce a trading profit
that will cover the remaining slippage between the Fix benchmark and the (e↵ective) rate at which the 200
million GBP were purchased.
Several aspects of the third trading strategy are particularly noteworthy. First, the strategy necessitates
trades to establish and close out the speculative position in addition to the trades necessary to fill the
fill-at-fix order. Consequently, there would be greater trading volume around the Fix if many dealer-banks
follow this strategy than is necessary to simply process the fill-at-fix orders across the market. Second, the
strategy requires an inclination on the part of dealer-banks to take speculative positions. Generally speaking,
dealer-banks will be more willing to take such positions the more representative they believe their fill-at-fix
orders are relative to others across the market. For if their orders are indeed representative, they provide
information on the aggregate order flow that must be processed by the market between 3:45 and 4:00 pm.
Consistent with large body of research, dealer-banks know that order flow is the dominant driver of spot rates
(away from scheduled data releases), so they will be willing to take a speculative position to benefit from
10 While
these are not legally binding guarantees, it is very rare for fill-at-fix orders not to filled at the Fix benchmark rate.
could also accept a limit order where price-contingency replaces the immediacy feature of the forex order.
11 Dealer-bank
8
the anticipated impact of order flow on future rates. Under these circumstances, the trades used by dealers
to initiate their speculative positions will be in the same direction as the trades they use to incrementally
fill the fill-at-fix orders – a trading pattern referred to as “front running”.
In sum, the economic relevance of the Fix arises from the fact that it is used in real-time valuation. This,
in turn, creates incentives for atypical forex trading activity around the 4:00 pm. There is a strong hedging
incentive for fund managers and derivative investors to submit fill-at-fix forex orders to dealer-banks before
3:45 pm, particularly at the end of the month. And, once these atypical forex orders are received, there are
strong incentives for dealer-banks to trade in a way that mitigates the risk inherent in filling the orders.
The key challenge in examining the behavior of the forex market around the Fix is understanding how this
trading activity is reflected in the behavior of spot rates.
1.2
Theoretical Background
The institutional features described above do not, in and of themselves, provide an explanation for the
behavior of spot rates around the Fix. The submission of fill-at-fix forex orders before 3:45 pm and their
implications for risk-mitigating trades by dealer-banks do not comprise a trading theory that can account
for the volatility and negative serial correlation in spot rate changes around the Fix found in the data. What
we require, instead, is an understanding of how the decisions by all market participants (i.e., dealer-banks
and others) give rise, in aggregate, to the unusual behavior of spot rates we observe. In short, we need a
model of forex trading that incorporates the institutional features described above and delivers equilibrium
spot rates with the same statistical characteristics as we find in the data.
The Portfolio Shifts (PS) model developed by Lyons (1997) and Evans and Lyons (2002) and extended
in Evans (2011) provides some useful insights into the behavior of spot rates around the Fix. The model
explains how the optimal trading decisions of a large number of dealer-banks drive the dynamics of spot
rates over the trading day. In particular, it describes how dealer-banks trade with one-another after they
have received and filled forex orders from investors (non-banks), and how resulting pattern of inter-dealer
trading is reflected in the behavior of spot rates.
The first insight arises from the characteristics of the model’s equilibrium. As in standard models,
equilibrium (bid and o↵er) spot rates clear markets. In the context of a forex trading model this means
that there must be willing counterparties to all currency trades. In addition, the spot rates at any point in
time support an ex ante efficient risk-sharing allocation across all market participants. Efficient risk-sharing
requires that the marginal utility from holding forex (either a single currency or a portfolio) is the same
across all market participants in every possible state of the world. This allocation is achieved at the end of
each trading day in the PS model because the spot rate reaches a level where the entire stock of forex is held
by (non-bank) investors. This aspect of the model’s equilibrium accords well with the fact that dealer-banks
do not hold substantial overnight forex positions. Risk-sharing also a↵ects the determination of spot rates
earlier in the trading day. Specifically, they adjust to levels consistent with market clearing and participants’
forecasts for the end-of-day rates conditioned on common information. This doesn’t mean that the intraday
spot rates necessarily follow a random walk. In fact they don’t in the PS model. In equilibrium there can be
predictable patterns in rates that lead market participants to take (di↵erent) speculative positions, so long
as in aggregate this speculative behavior is consistent with market clearing.
9
The relevance of these theoretical implications for the behavior of spot rates around the Fix is straightforward. When viewed from the perspective of the whole market, the hedging incentives to trade at the Fix
are likely to produce changes in the distribution of forex holdings across non-bank participants. Thus, from
the perspective of the PS model, trading around the Fix should establish a level for the spot rate at which
the post-fix distribution of forex holdings achieves an efficient risk-sharing allocation. To see what this would
mean in practice, consider the following examples.
Suppose that while individual dealer-banks receive positive and negative net fill-at-fix purchase orders for
USD against GBP, in aggregate the orders net to zero. Furthermore, for the sake of clarity, let us assume that
all dealer-banks hold their desired forex positions at 3:45 pm and that no other participants submit standard
forex trades around the Fix. Under these circumstances, the PS model implies that the Fix benchmark
will equal the (mid-point) of the bid and o↵er rates at 3:45 pm because those rates are consistent with an
efficient risk-sharing allocation of forex after the Fix. Dealer-banks are able to fill their fill-at-fix orders by
trading with each other at 4:00 pm without generating unwanted long or short positions, and post-fix forex
holdings of non-banks will be at desired level because spot rates remain unchanged between 3:45 and 4:00
pm. Moreover, in the absence of external factors generating further changes in the desired forex holdings of
non-banks, spot rates should remain at the level of the Fix for the remainder of the trading day.
Under other circumstances the aggregate imbalance in fill-at-fix orders will necessitate the establishment
of a equilibrium spot rate that di↵ers from the 3:45 pm rate. Now the fill-at-fix orders can only be filled
if dealer-banks as a group take either a long or short position, so the spot rates generated by inter-dealer
trading in the seconds around 4:00 pm do not represent the equilibrium rate at the end of the day’s trading.
Instead there must be an further change in the spot rate to a level at which dealers can find non-bank
participants with which they can trade away their unwanted long or short forex positions. The observed
behavior of spot rates around the Fix depends on the speed of this process. If it takes place within the one
minute Fix window, the benchmark will closely approximate the end-of-day equilibrium spot rate. In this
case there would be a significant pre-fix change in spot rates between 3:45 and 4:00 pm and an insignificant
post-fix change. Alternatively, if the process extends well beyond the end of the Fix window, there would
be significant pre- and post-fix spot rate changes.
In sum, the PS model provides an insight into why the Fix benchmark may be at a somewhat di↵erent
level than spot rates before and after 4:00 pm. Simply put, spot rates appear volatile around the Fix because
they are adjusting to a new distribution of desired forex holdings by non-banks participants.
The second important insight from the PS model concerns the trading behavior of dealers. In the model
dealer-banks use information contained in the forex orders they receive from non-bank investors to forecast
future movements in spot rates from which they establish speculative positions via their trades with other
dealer-banks. The forex orders received by individual dealer-banks have forecasting power because they
represent a noisy signal concerning the new distribution of desired forex holdings by non-bank investors that
the future equilibrium spot rate must accommodate. Importantly, the model shows that dealer-banks trade
in the same direction when establishing their speculative positions as the incoming forex orders they receive
from non-banks. So if a dealer-bank received a order to purchase GBP with USD, say, he would in turn
purchase GBP from other dealers to set up a long speculative position in the GBP in anticipation of a rise in
the USD/GDP spot rate. This trading behavior does not constitute front running because the dealer-bank
10
fills the investor’s order before establishing the speculative position. Nevertheless, the dealer-bank would
want to trade in exactly the same manner if instead the investor’s order was filled at a later point in time.
In this sense the PS model provides a rationale for why dealer-banks would establish speculative positions
via trades that would appear to front run fill-at-fix forex orders. Front running arises as an optimal trading
strategy by dealer-banks who understand that the fill-at-fix orders contain (imprecise) information about
the future level of the spot rate consistent with an efficient risk-sharing allocation of forex holdings across
market participants at the end of the trading day.
Four key points arise from this insight. First, the presence of front running is not in and of itself an
indicator of Pareto inefficiency in forex trading. It could be part of dealer-banks’ optimal trading strategies
in the equilibrium of a forex trading model where the spot rate achieves a level consistent with an efficient
risk-sharing allocation by the end of the trading day. Second, the presence of front running by dealer-banks
need not a↵ect the behavior of spot rates. Limiting the size of dealer-banks speculative positions in the PS
model would not change the behavior of equilibrium spot rates during the day, but it would make acting
as a dealer-bank less attractive to potential market participants. Third, the size of dealer-banks speculative
positions (and hence the degree of front running) depend critically on the perceived precision of their spot
rate forecasts. Risk-averse dealer-banks understand that their forecasts are based on imprecise inferences
about the new distribution of desired forex holdings across all non-bank participants, and so choose the
size of their speculative positions to balance expected profits against the risk of actual losses. Under these
circumstances, information about the orders received by other dealer-banks would be economically valuable
to any individual dealer-bank because it would improve the precision of its spot rate forecasts and reduce
the risk associated with taking a particular position.
The forth and final point concerns the relation between front running and serial correlation in spot rate
changes. In the PS model, spot rates jump directly to their end-of-day equilibrium level immediately after
dealer-banks trade to establish their speculative positions. Thereafter, they remain at the same level even
as the speculative positions are unwound and any undesired dealer-banks forex holdings are traded away to
non-banks. Consequently there is no serial correlation in spot rate changes between the time when individual
dealer-banks receive forex orders from investors and the end of the daily trading. This fact undermines the
idea that the existence of front running must lead to negative serial correlation in spot rate changes. It also
means that the PS model cannot provide a complete explanation for the behavior of spot rates around the
Fix.
Could front running produce a negative serial correlation in equilibrium spot rate changes in another
trading model? Possibly, but the model would have to limit the ability or inclination of market participants
to exploit the predictability in spot rate movements. In the presence of negative serial correlation all
participants will generally have an incentive to take long (short) speculative positions follow a fall (rise) in
rates, so it will be impossible to find the counterparties necessary for the trades that initiate the positions
unless speculative trading is limited to a subset of market participants. Alternatively, some participants
must have a strong, overriding incentive to act as counterparties to the speculative trades of others. Section
7 considers further the incentive to take speculative positions that exploit the negative serial correlation in
spot rate changes around the Fix.
In summary, the PS model of forex trading provides a number of insights into the possible factors driving
11
the behavior of spot rates around the Fix. In particular, it provides insights into the source of spot rate
volatility and the possible presence of front running by dealer-banks. That said, the PS model (and other
forex trading models) does not provide an “o↵-the-self” explanation for the negative correlation between
pre- and post-fix spot rate changes that appears to be a prominent feature of the end-of-month data - a
point I return to in Section 7 below.
2
Data and Statistical Methods
2.1
Data Sources
I use data from two sources. The daily Fix benchmarks are taken from Datastream. The intraday spot rate
data comes from Gain Capital, a provider of electronic Forex trade data and transaction services, and the
parent company for the retail trading portal Forex.com. Their data archive includes tick-by-tick bid and o↵er
rates for a wide range of currencies, some starting as far back as 2000. In this study I focus on the spot rates
for 21 currency pairs: the four majors involving the U.S. Dollar (USD/EUR, CHF/USD, USD/GBP and
JPY/USD) and 17 further rates that use either the Euro, Pound or Dollar as the base currency. These rates
are listed in column (i) of Table 1. Columns (ii) and (iii) report the span and scope of the tick-by-tick data
for each rate. For 11 currency pairs I use a decade of tick-by-tick bid and o↵er rates starting at midnight on
December 31 st., 2003. Continuous data is not available for the other currency pairs in 2004 – 2007 so I use
tick-by-tick rates starting after midnight on December 31st 2007, when continuous data becomes available.
The data samples for all the currency pairs end at midnight on December 31 st., 2013. As column (iii) shows,
the time series for each currency pair contains tens of millions of data points. Each series contains a date
and time stamp, where time is recorded to the nearest 1/100 of a second, and a bid and o↵er rate. Unlike
standard time series, the time between observations is irregular, ranging from a few minutes to a hundredth
of a second.
Gain Capital aggregates data from more than 20 banks and brokerages in the Forex market to construct
the bid and o↵er rates for each currency pair. To gauge how accurately these data represent rates across the
Forex market, Gain provides a comparison of the mid-point between its bid and ask rates with the mid-point
for the best tradable bid and ask rates aggregated from 150 market participants by an independent firm,
Interactive Data Corporation GTIS. These comparisons (available on line at http://www.forex.com/pricingcomparison.html) show very small di↵erences between the two mid-point series in current data, typically less
than one pip.12
As a further check on the accuracy of the Gain data, I compared the mid-points from the tick-by-tick
data with the 4:00 pm Fix benchmarks on each trading day in the sample. Recall that the Fix benchmarks
are computed as the mid point of the median bid and ask rates across multiple transactions in one minute
window that starts 30 seconds before 4:00 pm. For comparison I computed an analogous mid-point from
the median of the bid and ask rate data on every trading day covered by each currency pair. Di↵erences
12 In the Forex market a “pip” typically refers to the fourth decimal place in a spot rate, i.e., the di↵erence between a
EUR/USD rate of 1.3745 and 1.3743 is three pips. Rates involving the JPY are an exception to this convention, where a pip
refers to the second decimal; e.g. there is a two pip di↵erence between the JPY/USD rates of 107.42 and 107.44. In my analysis
I report di↵erences between rates in basis points (i.e., 1/100 of a percent) rather than pips to facilitate comparisons across
di↵erent currency pairs.
12
between this mid-point and the Fix represent the tracking error of the Gain data relative to the rates used
to determine the Fix.13
Table 1 reports the percentiles of the tracking-error distribution, measured in basis points relative to the
Fix benchmark, for each of the currency pairs I study. Since the behavior of spot rates around the Fix on the
last trading day in each month have been subject to particular scrutiny by the financial press, I separate the
tracking errors on these days from the errors on other trading days and report percentiles for both the intraand end-of-month distributions. Table 1 shows that the tracking errors in the Gain data are typically very
small. The center panel of the table shows that the vast majority of intra-month tracking errors are within
±2 basis points. This represents a high level of accuracy. For perspective, column (xii) reports the average
spread between the bid and ask rates for each currency pair between 3:00 and 5:00 pm GMT. Clearly, most
of the tracking-error distributions lie within these average spreads. The distributions for the end-of-month
tracking errors are a little more dispersed: the 5’th. and 95’th. percentiles reported in columns (ix) and (xi)
are larger (in absolute value) than their counterparts in the intra-month distributions (see columns (v) and
(vii)). That said, the vast majority of the end-of-month tracking errors are still very small, both in absolute
terms and relative to the average spreads.
Table 1 also reports the number of trading days used to compute the tracking-error distributions in
columns (iv) and (viii). In my analysis below I only use the Gain tick-by-tick data on days where the timestamps for each bid and ask rate can be exactly matched to GMT. Unfortunately, this is not always possible.
There are days where the bid and ask rates with time-stamps that should correspond to 4:00 pm are clearly
far from the Fix, so there must be a recording error in the Gain archive. I do not use any of the Gain data
on these days. The di↵erent trading day numbers reported in columns (iv) and (viii) reflect the e↵ects of
this data verification process as well as di↵erences in the data spans across currency pairs.
In summary, the statistics in Table1 show that once the accuracy of the time-stamps in the Gain data
has been verified, the tick-by-tick rates around the 4:00 pm very closely match the rates used in computing
the actual Fix. Importantly, the tracking errors documented here are much smaller in magnitude than the
changes in rates we will examine in the periods before and after the 4:00 pm, so the Gain data provides an
accurate measure of how forex rates behave across the market around the Fix.
2.2
Statistical Methods
The statistical methods I use below are chosen to highlight how the behavior of spot rates around the end-ofmonth Fixes di↵er from their behavior around intra-month Fixes, and other times. To accommodate the fact
that the time series for intraday rates are irregularly spaced (i.e., the time between consecutive observations
di↵ers from observation to observation), I use a set of “observation windows” that define market events
in clock time around the 4:00 pm. The set of observation windows are shown in Table 2. They range in
duration from 11 hours starting at 7:00 am and ending at 6:00 pm, to just two minutes between 3:59 and
4:01 pm. For each window on every trading day with reliable Gain data I compute statistics that summarize
the behavior of the mid-point rate (i.e., the average of the bid and o↵er rates) within the window. These
statistics include the first and last rates, the maximum and minimum rates.
13 All
calculations are undertaken using Matlab.
13
14
CHF/EUR
JPY/EUR
NOK/EUR
NZD/EUR
SEK/EUR
AUS/GBP
CAD/GBP
CHF/GBP
EUR/GBP
JPY/GBP
NZD/GBP
AUS/USD
CAD/USD
DKK/USD
NOK/USD
SEK/USD
SGD/USD
B:
C:
D:
2004-13
2004-13
2008-13
2008-13
2008-13
2008-13
2008-13
2008-13
2004-13
2004-13
2004-13
2008-13
2004-13
2004-13
2008-13
2008-13
2008-13
2004-13
2004-13
2004-13
2004-13
49.016
36.163
66.719
55.350
58.296
10.567
68.169
57.455
83.686
41.643
88.578
58.216
37.858
78.813
15.780
56.633
17.424
55.370
51.966
38.931
60.859
(iii)
(millions)
Prices
2398
2404
1305
1306
1297
1200
1476
1478
2417
2339
2418
1409
2373
2421
1291
1414
1288
2420
2258
2204
2421
(iv)
Number
-1.601
-1.461
-0.696
-1.696
-1.811
-1.440
-1.209
-1.180
-1.476
-1.651
-1.564
-2.197
-1.144
-1.538
-1.710
-2.211
-1.643
-1.113
-1.510
-1.268
-1.087
(v)
0.144
0.138
0.081
0.071
0.053
0.000
0.225
0.293
0.087
0.114
0.020
0.089
0.000
0.021
0.008
0.079
-0.010
0.055
0.060
0.140
0.083
(vi)
2.283
1.864
0.825
2.152
1.792
1.517
1.730
1.777
1.634
2.176
1.582
2.350
1.145
1.542
2.015
2.381
1.551
1.209
1.761
1.790
1.200
(vii)
Tracking Error Distribution
Percentiles (basis points)
5%
50%
95%
116
116
59
62
59
61
69
71
116
115
116
67
116
117
62
68
59
117
106
104
116
(viii)
Number
-2.373
-1.909
-0.779
-3.983
-2.783
-1.980
-2.695
-1.676
-3.412
-2.302
-2.612
-2.529
-2.135
-4.939
-2.624
-2.927
-2.336
-1.232
-2.462
-3.241
-1.258
(ix)
-0.117
0.256
0.115
0.602
0.101
0.168
0.544
0.371
0.020
0.157
0.137
0.397
0.000
0.090
0.252
0.284
-0.089
0.109
0.058
0.207
0.051
(x)
2.053
2.821
0.996
3.999
2.067
1.982
4.016
2.578
2.013
2.523
2.378
5.046
1.243
2.822
3.829
3.844
1.980
1.771
2.345
2.608
2.566
(xi)
Tracking Error Distribution
Percentiles (basis points)
5%
50%
95%
End-of-Month Trading Days
3.171
3.576
1.244
4.738
4.048
3.671
4.773
4.841
4.152
3.208
4.090
9.738
2.160
2.622
4.449
7.018
3.584
1.708
3.477
2.771
2.285
(xii)
Average
Spread
(basis points)
Notes: Columns (i) - (iii) show the data span and the number of quotes (in millions) for each of the currency pairs in the data set. Columns (iv) and (viii)
report the number of intra-month and end-of-month trading days for which there are intraday quotes, respectively. Quote errors on each day are defined as the
di↵erence between the mid-point of the average bid and ask quotes computed over a 30 second window centered on 4:00 pm and the Fix benchmark. Quote
errors are expressed in basis points. Columns (v) - (vii) and (ix) - (xi) show the 5th., 50th. and 95th. percentiles of the quote error distribution computed on
all intra-month and end-of-month trading days. Column (xii) reports the average spread (in basis points) between the bid and ask quotes between 3:00 and
5:00 pm.
EUR/USD
CHF/USD
JPY/USD
USD/GBP
(ii)
(i)
A:
Data Span
FX Rate
Intra Month Trading Days
Table 1: Data Characteristics
Table 2: Observation Windows
Window
Start Time
End Time
Duration
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
7:00
3:00
3:30
3:45
3:50
3:55
3:56
3:57
3:58
3:59
6:00
5:00
4:30
4:15
4:10
4:05
4:04
4:03
4:02
4:01
11 hrs
2 hrs
1 hr
30 mins
20 mins
10 mins
8 mins
6 mins
4 mins
2 mins
am
pm
pm
pm
pm
pm
pm
pm
pm
pm
pm
pm
pm
pm
pm
pm
pm
pm
pm
pm
I also use the Gain data to constructed empirical distributions for intraday spot rate dynamics away from
the Fix. To build these distributions I pick a random starting time between 7:00 am and 6:00 pm on any
day from the span of the intraday time series for a specific rate. I then use this time as the starting time for
nine observation windows that range in duration from two hours to two minutes. These randomly selected
windows correspond to windows (ii) to (x) in Table 2. If any of the randomly selected windows cover the
Fix or the release of U.S. macro data at 8:30 am EST, I discard the starting time. If not, I compute and
record the same series of statistics for each of the nine windows (again using mid-point rates). This process
is repeated 10,000 times to build up the empirical distribution of the rate statistics away from the Fix. It
is important to exclude observation windows that cover the scheduled releases of U.S. macro data when
constructing these empirical distributions because the releases are often accompanied by large rate changes.
These empirical distributions provide a benchmark to quantify di↵erences between the behavior of spot rates
around the Fix and other periods of normal trading activity.
In the next 4 sections I examine the behavior of rates around the Fix. To begin I take a macro perspective.
Fix benchmarks are routinely used to identify the daily spot rates from which the time series of exchange
rates over months, years and decades are constructed, yet they are derived from spot rates contained in a
very narrow window of daily trading activity. Section 3 examines the implications of this limitation. Next,
in Section 4, I describe the behavior of spot rates under normal trading conditions. This analysis establishes
empirical metrics that are used when I study the behavior of rates immediately before and after 4:00 pm in
Sections 5 and 6, respectively.
3
Daily Trading Ranges and the Fix
The forex market operates continuously, without any set opening or closing times, but in reality most trading
is heavily concentrated on weekdays between approximately 7:00 am and 6:00 pm GMT. In contrast, the
15
spot rates used to compute the Fix come from a tiny window of daily trading activity: 30 seconds either side
of 4:00 pm. Consequently, each day’s Fix provides limited information on the rates at which currencies trade
throughout the trading day. Here I examine the implications of this limitation when studying the behavior
of spot rates over days, months and longer horizons.
Figure 2: Major Currency Fixes with Daily Trading Range
EUR/USD
CHF/USD
1.6
1.4
1.55
1.3
1.5
1.2
1.45
1.1
1.4
1.35
1
1.3
0.9
1.25
0.8
1.2
1.15
0.7
05
06
07
08
09
10
11
12
13
05
06
07
08
09
10
11
12
13
10
11
12
13
USD/GBP
JPY/USD
125
2.1
120
2
115
1.9
110
105
1.8
100
1.7
95
90
1.6
85
1.5
80
75
1.4
05
06
07
08
09
10
11
12
13
05
06
07
08
09
Notes: Time series for the Fix at the end of each month with upper and lower limits of daily trading range.
The Fix benchmarks are routinely used as daily rates when constructing time series for spot exchange
rates over days, months or years. Figure 2 plots monthly time series for the spot rates of the four major
currency pairs using the end-of-month Fixes between the end of 2003 and 2013. The plots also show the
upper and lower limits for (mid-point) rates between 7:00 am and 6:00 pm GMT on the last trading day
of each month. As these plots clearly indicate, the low frequency variations in the level of each spot rate
(between one and five years in duration) are orders of magnitude larger than the daily rate ranges. Thus
the low frequency time series characteristics of spot rates appear robust to the use of the Fix to identify the
end-of month rates. One way to visualize this is to imagine alternative plots where the end-of-month rate
is pinned down by a randomly chosen point within the daily trading range. The plots would undoubtedly
look a little di↵erent from one month to the next, but they would still closely track the long swings shown
in Figure 2. The Appendix contains analogous plots for the other 17 exchange rates that exhibit the same
16
features as the plots in Figure 2. In sum, therefore, the use of the Fix to identify the daily spot rate does
not materially a↵ect how we view the evolution of exchange-rate levels over long horizons.
Figure 3: Daily Trading Ranges around the Fix
EUR/USD
CHF/USD
300
200
250
150
200
100
150
50
100
0
50
−50
0
−100
−50
−150
−100
−200
−150
−200
−250
05
06
07
08
09
10
11
12
13
05
06
07
08
JPY/USD
09
10
11
12
13
10
11
12
13
USD/GBP
200
250
150
200
100
150
50
100
0
50
−50
0
−100
−50
−150
−100
−200
−150
05
06
07
08
09
10
11
12
13
05
06
07
08
09
Notes: Each panel plots the daily price range at the end of each month as a band around the Fix price in basis points. The upper and lower edges of the band are equal to
(ln Pth ln Ptf )10000 and (ln Ptf ln Ptl )10000, respectively; where Ptf is the Fix price, Pth is the maximum price and Ptl is the minimum price between 7:00 am and 6:00 pm
GMT on day t.
While daily spot rate ranges are small compared to the long-term swings in the level of rates, they are
nevertheless sizable. Figure 3 illustrates this point for the major currency pairs. Here I plot the daily range
at the end of each month as a band around the Fix in basis points. Thus the upper and lower edges of the
band are equal to 10000(ln Stmax
ln Stf ix ) and
10000(ln Stf ix
ln Stmin ), respectively; where Stf ix is the
Fix benchmark, Stmax is the maximum rate and Stmin is the minimum rate between 7:00 am and 6:00 pm
on day t. As the plots clearly show, the ranges are sometimes as large as a couple of hundred basis points
(particularly during the 2008-2009 financial crisis), and are often at least a hundred basis points. Notice,
also, that the bands are rarely symmetric around zero because the Fix is often far from the center of the
daily range; a point I shall return to below. As in Figure 2, these plots are representative of the bands for
the other currency pairs shown in the Appendix.
One way to judge the economic significance of the daily spot rate ranges is to compare them against
prior changes in the Fix over di↵erent horizons. For this purpose, I compute the range-to-change ratio
17
Rn = (ln Stmax
ln Stmin )/| ln Stf ix
ln Stf ixn | at the end of each month for horizons n of one month, one
quarter and one year. Rn is just the ratio of the daily range (in percent) on day t to the absolute value of
the percentage change in the Fix from day t
n to day t. Table 3 reports the 50th. and 90th. percentiles
of the empirical distributions for Rn at three horizons for all the currency pairs. As the table shows, for all
the currency pairs both the 50’th. and 90’th. percentiles fall as the horizon rises from one month to one
year. This is indicative of the leftward shift in the Rn distributions as n rises, which is not at all surprising.
What is surprising are the size of ratios. To understand why, suppose an investor initiated a position at the
Fix at the end of last month that was closed out at today’s Fix, a month later, with a 1 percent return. If
Rn = 0.5 today, and the investor had the discretion to close out the position at any time between 7:00 am
and 6:00 pm, he could have potentially achieved a return as large as 1.5 percent or as small as 0.5 percent,
depending on where today’s Fix was set relative to the daily range. In this sense the median values for Rn
imply that monthly and quarterly returns computed from Fix benchmarks are “typically” rather imprecise
measures of the return an investor might have received had they initiated and/or closed their positions away
from the Fix on the same days. Moreover, on at least ten percent of the days covered by the sample, returns
computed from the Fix could have been very imprecise. As the right hand columns of Table 3 show, the
90’th. percentiles of the Rn distributions are in many cases above one. In these instances it is possible that
the return an investor received on a position initiated at the Fix but closed away from the Fix would have
a di↵erent sign from one closed at the Fix.
The results in Table 3 make clear that forex returns computed over macro-relevant horizons are sensitive
to the time of day that positions are initiated and closed. Unless investors are known to only execute their
forex trades at the Fix, conventional measures of returns on forex positions that use the Fix as the daily
exchange rate are potentially very imprecise measures of the returns actual investors received from positions
initiated and closed on the same days. Of course the exact level of imprecision depends on far the rates
received by the investor on their transactions to initiate and close the position di↵er from the Fix. These
calculations require trading data on individual investors. In contrast, most of the research literature on the
carry trade, forward premium puzzle, and international portfolio diversification implicitly assumes that the
ability to trade away from the Fix has no material a↵ect on Forex returns over macro horizons. At the very
least, the results in Table 3 cast some doubt on this assumption.
The results in Table 3 also provide a perspective on why so many forex trades are executed at the Fix.
When an investor sells a foreign currency denominated security (e.g. a stock or bond) held in a custodian
account, the proceeds from the sale are used to purchases domestic currency that is credited to the investor’s
account. The results in the Table 3 show that the (domestic currency) return the investor ultimately receives
could be materially a↵ected if the custodian has discretion to choose the rate for the forex trade within the
range on the day the security is sold. Indeed the choice of rate for such forex trades has been the subject
of litigation between institutional investors (mutual and pension funds) and custodial banks.14 One way to
avoid such litigation is to eliminate discretion over the rate used in custodial forex trades by specifying that
they are executed at the Fix. This arrangement increases the level of transparency in custodial trades for
institutional investors and also produces a flow of orders into the forex market to execute trades at the Fix.
14 See: Louisiana Municipal Police Employees’ Retirement System et al v. JPMorgan Chase & Co et al, U.S. District Court,
Southern District of New York, No. 12-06659; and Bank of New York Mellon Corp Forex Transactions Litigation in the same
court, No. 12-md-02335.
18
Table 3: Range-to-Change Ratios
Rn = (ln Stmax
horizons n
ln Stmin )/| ln Stf ix
50th. percentile
1 month 1 quarter 1 year
ln Stf ixn |
1 month
90th. percentile
1 quarter
1 year
(i)
(ii)
(iii)
(iv)
(v)
(vi)
A:
EUR/USD
CHF/USD
JPY/USD
USD/GBP
Average
0.430
0.460
0.369
0.536
0.449
0.222
0.224
0.207
0.312
0.241
0.107
0.203
0.084
0.168
0.141
2.016
2.527
2.230
5.184
2.989
1.646
1.518
1.420
2.071
1.664
0.611
1.418
0.372
0.989
0.848
B:
CHF/EUR
JPY/EUR
NOK/EUR
NZD/EUR
SEK/EUR
Average
0.547
0.367
0.560
0.405
0.478
0.472
0.288
0.214
0.303
0.192
0.290
0.257
0.123
0.094
0.131
0.114
0.111
0.115
4.258
3.460
2.104
1.561
2.075
2.691
1.969
1.144
1.403
0.907
1.772
1.439
0.366
0.596
0.502
0.738
0.688
0.578
C:
AUD/GBP
CAD/GBP
CHF/GBP
GBP/EUR
JPY/GBP
NZD/GBP
Average
0.372
0.529
0.451
0.493
0.383
0.416
0.441
0.177
0.419
0.278
0.264
0.227
0.238
0.267
0.116
0.235
0.123
0.172
0.096
0.142
0.147
2.991
3.433
2.616
3.896
1.181
1.443
2.593
1.115
1.777
1.567
1.832
0.903
1.912
1.518
0.982
1.164
0.558
0.980
0.959
1.390
1.005
D:
AUD/USD
CAD/USD
DKK/USD
NOK/USD
SEK/USD
SGD/USD
Average
0.355
0.469
0.432
0.470
0.491
0.304
0.420
0.236
0.284
0.214
0.275
0.304
0.190
0.251
0.096
0.136
0.121
0.215
0.195
0.099
0.144
1.457
2.544
1.861
2.597
2.286
2.084
2.138
1.319
1.364
1.163
1.141
4.565
0.861
1.735
0.385
0.694
0.534
1.570
1.161
0.373
0.786
Notes: The table reports percentiles of the empirical Rn distributions for each of the exchange rates listed on the left.
Empirical distributions are constructed from the values for Rn computed at the end of each month for which reliable
intraday rate data is available.
Table 4 reports statistical results that compliment the visual evidence in Figure 3 on the relation between
the daily spot rate range and the Fix at the end of each month. The table provides information on the intraday
rate ranges between 7:00 am and 6:00 pm, 3:00 and 5:00 pm, and between 3:30 and 4:30 pm on every day for
which there is reliable data for each currency pair. Columns (i) and (ii) report the 50th. and 90th. percentiles
of the empirical distribution for the range expressed in basis points; i.e., 10000(ln(S max )
ln(S min )) where
S max and S min are the highest and lowest (mid-point) rates within the range. The tail probabilities in
columns (iii) and (iv) compare the Fix to the range on each day. Specifically, column (iii) reports the
19
fraction of days on which the ratio (S f ix
S min )/(S max
S min ) is either below 0.1 or above 0.9, while
column (iv) reports fraction on which the ratio is either below 0.05 or above 0.95.
An inspection of the statistics in Table 4 reveals several noteworthy features. First, there is remarkable
similarity in the empirical range distributions across currency pairs. Column (i) shows that typical spot rate
ranges (represented by the 50th. percentiles) from 7:00 am to 6:00 pm are between 70 and 80 basis points,
fall to around 30 points between 3:00 and 5:00 pm, and are on average a little above 20 points between 3:30
and 4:30 pm. The 90th. percentiles for the range distributions are also very similar across most currency
pairs, and are roughly twice the size of the 50th percentiles. Four currency pairs prove exceptions to this
pattern: Distributions for the CHF/EUR and SGD/USD are shifted more to the left, while those for the
NOK/USD and SEK/USD are shifted more to the right.
The second noteworthy feature concerns the e↵ect of time on the range distributions. As one would
expect, the distributions shift leftward and become more compact as the ranges are computed over shorter
time windows. Notice, however, that the statistics in panel III are based from just one hour of trading
activity whereas those in panel I come from 11 hours. If the sequence of intraday rates followed a random
p
walk with a constant variance, the percentiles in panel I should be 11 ' 0.33 times their counterparts in
panel III. The table shows that this is approximately the case. This is surprising because the statistics in
panel I encompass periods during which macro data are routinely released, whereas those in panel III come
from the hour of trading around the Fix where releases do not occur. The factors a↵ecting rates around the
Fix appear comparable in their e↵ects on the range of rates as the release of macro data. This is one piece
of evidence documenting the atypical behavior of spot rates around the Fix.
The third feature concerns the tail probabilities reported in columns (iii) and (iv). As the table clearly
shows, the Fix appears close to the edges of the price ranges far more often that we would expect if it were
merely a randomly chosen point from the range. For a perspective, consider the position of an investor who
is committed to undertaking a forex trade on a particular day and must decide whether to execute the trade
via the submission of a standard (market or limit) order at a time close to 4:00 pm, or via the submission of a
fill-at-fix order. The tail probabilities in panels II and III imply that the investor faces more rate uncertainty
in orders filled at the Fix than from standard trades executed at a random time around the fix.
In summary, the results above show that the Fix provides limited information about the rates used in
the execution forex trades on any particular day. The Fix is computed as an average of rates in a narrow
one-minute window that cannot adequately represent the fully range of spot rates at which trades take place
over the trading day. As a consequence, investors initiating and closing positions away from the Fix are
quite likely to achieve returns over days, weeks and longer, that di↵er significantly from those computed over
the same horizons using the Fix. Furthermore, the Fix should not be viewed as representing a randomly
chosen spot rate from the intraday range on a particular day. Across all the currency pairs, the incidence of
Fix benchmarks near the edge of the intraday spot rate range is far higher than would be the incidence of
randomly chosen rates.
20
21
CHF/EUR
JPY/EUR
NOK/EUR
NZD/EUR
SEK/EUR
Average
AUS/GBP
CAD/GBP
CHF/GBP
EUR/GBP
JPY/GBP
NZD/GBP
Average
AUS/USD
CAD/USD
DKK/USD
NOK/USD
SEK/USD
SGD/USD
Average
B: EUR
C: GBP
D: USD
78.218
74.574
80.139
105.594
110.334
36.736
80.932
79.906
82.238
66.053
57.296
81.113
86.413
75.503
32.996
79.185
61.523
82.317
65.110
64.226
73.049
79.157
66.341
68.880
71.857
161.009
137.799
146.297
197.482
209.301
67.820
153.285
155.525
153.473
133.963
112.261
165.301
161.864
147.064
90.981
163.978
121.154
151.633
129.011
131.352
133.130
142.709
120.889
129.649
131.594
(ii)
0.329
0.284
0.304
0.311
0.299
0.313
0.307
0.294
0.288
0.286
0.248
0.293
0.297
0.284
0.340
0.299
0.272
0.298
0.260
0.294
0.304
0.321
0.304
0.279
0.302
(iii)
0.218
0.181
0.216
0.198
0.192
0.185
0.198
0.202
0.176
0.190
0.154
0.177
0.187
0.181
0.222
0.192
0.163
0.204
0.153
0.187
0.210
0.216
0.197
0.177
0.200
(iv)
37.792
35.149
37.026
50.165
51.952
16.850
38.156
36.362
38.710
28.722
23.686
34.818
41.723
34.003
15.306
35.100
28.879
38.680
29.969
29.587
32.923
35.972
29.651
29.767
32.078
(i)
81.713
70.505
70.223
94.826
98.312
31.507
74.515
74.123
76.689
59.177
46.904
75.997
82.508
69.233
41.911
74.383
55.579
76.462
57.276
61.122
64.408
68.733
59.880
59.391
63.103
(ii)
0.368
0.329
0.410
0.347
0.350
0.344
0.358
0.361
0.315
0.357
0.327
0.347
0.335
0.340
0.334
0.363
0.277
0.340
0.283
0.319
0.408
0.396
0.373
0.357
0.384
(iii)
0.227
0.202
0.267
0.220
0.213
0.225
0.226
0.230
0.203
0.220
0.192
0.232
0.202
0.213
0.208
0.235
0.167
0.201
0.174
0.197
0.270
0.253
0.243
0.228
0.248
(iv)
II: 3:00-5:00 pm GMT
Range Distribution Tail Probabilities
50%
90%
20%
10%
27.054
24.847
25.234
35.640
36.553
11.549
26.813
26.683
27.846
20.951
17.273
24.917
30.252
24.654
11.164
24.073
20.754
28.393
22.076
21.292
22.312
24.611
20.715
20.757
22.099
(i)
56.649
48.947
49.993
67.861
70.112
23.386
52.825
57.060
56.795
43.033
34.743
54.822
63.275
51.621
30.955
52.200
41.480
57.282
41.735
44.730
44.659
48.101
39.984
42.069
43.703
(ii)
0.330
0.303
0.398
0.330
0.325
0.313
0.333
0.338
0.305
0.333
0.302
0.325
0.298
0.317
0.315
0.364
0.246
0.309
0.250
0.297
0.392
0.359
0.346
0.338
0.359
(iii)
0.196
0.186
0.263
0.202
0.203
0.189
0.206
0.205
0.212
0.209
0.173
0.212
0.197
0.201
0.184
0.222
0.156
0.194
0.166
0.185
0.251
0.232
0.234
0.213
0.232
(iv)
III: 3:30-4:30 pm GMT
Range Distribution Tail Probabilities
50%
90%
20%
10%
Notes: Columns (i) and (ii) report the 50th. and 90th. percentiles from the empirical distribution of the trading range (identified in the header of each panel)
expressed in basis points; i.e., (ln(S max ) ln(S min ))10000 where S max and S min are the highest and lowest mid-point rates within the range. Column (iii) reports
the fraction of days in the sample that the ratio (S f ix S min )/(S max S min ) is either below 0.1 or above 0.9. Column (iv) reports the fraction of the days when
the ratio is either below 0.05 or above 0.95.
EUR/USD
CHF/USD
JPY/USD
USD/GBP
Average
A: Majors
(i)
I: 7:00 am -6:00 pm GMT
Range Distribution Tail Probabilities
50%
90%
20%
10%
Table 4: Trading Ranges and the Fix
4
Spot Rate Dynamics Away from the Fix
In this section I examine the behavior of intraday spot rate dynamics away from the Fix. Table 5 reports
statistics for the distribution of spot rate changes over horizons of five, fifteen, and thirty minutes. These
statistics are computed from an empirical distribution of 10000 observations chosen at random times (away
from the Fix) from the time series of intraday (mid-point) rates, {St }, for each currency pair (as described in
Section 2.2). Columns (iii) - (vii) report statistics for the distribution of changes in the log rates expressed
in basis points per minute, i.e.,
h
st ⌘ (ln(St+h )
ln(St )) ⇤ 10000/h for horizons h = {5, 15, 60} minutes.
Columns (viii) and (ix) report the first-order autocorrelation in
h
st (i.e. corr(
h
st+h ,
h
st )) and the
p-value for the null of a zero autocorrelation, respectively. Column (x) reports the Kolmogorov-Smirnov
h
(KS) test for the null that the two conditional distributions f (
are the same.
15
The p-value for the test is shown in column (xi).
st+h |
h
st > 0) and f (
h
st+h |
h
st  0)
As Table 5 shows, the rate-change distributions have several common characteristics across all the currency pairs. First, the dispersion in the rate-change distributions decline as the horizon rises. Columns (iii)
and (iv) show that the absolute values for the 5th. and 95th. percentiles of the distributions fall as the
horizon rise from five to 30 minutes. The change in dispersion is also reflected by the standard deviations
shown in column (v), which fall as the horizon rises. Second, all the rate-change distributions are strongly
leptokurtic. As column (vii) shows, the kurtosis statistics across all the currency pairs are large; much larger
than the value of three for the implied by the normal distribution. These statistics indicate that atypically
large changes in rates occur quite frequently away from the Fix and scheduled macro news releases.
The third feature concerns temporal dependence between rate changes.
rate changes display some small degree of autocorrelation.
Column (viii) shows that
Across currency pairs, the autocorrelation
is generally negative. This fact accounts for the declining dispersion of the rate-change distributions as
the horizon rises, noted above. Although small in (absolute) value, the statistics in column (ix) indicate that many of the estimated autocorrelation coefficients are statistical signifiant at standard levels.
There is also evidence of temporal dependence from the KS tests reported in column (ix). Under the null
of temporal independence, future changes in rates should not depend on the sign of past changes, i.e.,
f(
h
st+h |
h
st > 0) = f (
h
st+h |
h
st  0). As column (x) shows, this null can easily be rejected at
standard levels of significance for most currency pairs and horizons h.
15 Two versions of the KS test can be found in the statistics literature. The one-sample KS test is a nonparametric test of the
null hypothesis that the population cdf of the data is equal to the hypothesized cdf. The two-sample KS test is a nonparametric
hypothesis test of the null that the data in two samples are from the same continuous distribution. Here I compute the twosample KS test which uses the maximum absolute
di↵erence⌘ between the cdfs of the distributions of the two data samples.
⇣
The test statistic is computed as D = maxx |F̂1 (x) F̂2 (x)| where F̂1 (x) is the proportion of the first data sample less than
or equal to x, and F̂2 (x) is the proportion of the second data sample less than or equal to x. The KS test and its asymptotic
p-value are computed with the Matlab “kstest2” function.
22
Table 5: Spot Rate Dynamics
Spot Rate Changes (bps per minute)
A:
B:
C:
Temporal Dependence
horizon
5%
95%
std
skew
kurtosis
Autocorrelation
Independence
p-value
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
(xi)
EUR/USD
5
15
30
-1.345
-0.730
-0.468
1.378
0.729
0.468
0.886
0.466
0.302
-0.033
-0.122
0.057
8.777
7.694
9.717
-0.018
-0.007
0.025
(0.137)
(0.587)
(0.037)
0.055
0.047
0.047
(0.000)
(0.002)
(0.001)
CHF/USD
5
15
30
-1.481
-0.774
-0.510
1.532
0.787
0.492
0.968
0.511
0.318
-0.166
-0.090
-0.235
11.873
8.259
8.301
-0.021
-0.036
0.045
(0.097)
(0.005)
(0.000)
0.051
0.046
0.051
(0.001)
(0.003)
(0.001)
JPY/USD
5
15
30
-1.259
-0.657
-0.421
1.265
0.672
0.413
0.818
0.429
0.276
-0.009
0.310
0.198
8.457
8.110
9.298
-0.044
-0.047
0.033
(0.001)
(0.000)
(0.007)
0.049
0.055
0.050
(0.002)
(0.000)
(0.001)
USD/GBP
5
15
30
-1.317
-0.717
-0.460
1.338
0.711
0.473
0.915
0.501
0.329
0.285
-0.421
-0.633
12.967
20.581
28.025
-0.041
0.028
-0.049
(0.001)
(0.024)
(0.000)
0.043
0.026
0.047
(0.006)
(0.251)
(0.001)
CHF/EUR
5
15
30
-0.818
-0.464
-0.301
0.889
0.463
0.282
0.630
0.335
0.212
0.213
0.429
0.465
33.326
26.405
23.065
-0.046
-0.004
-0.010
(0.000)
(0.718)
(0.416)
0.072
0.057
0.047
(0.000)
(0.000)
(0.002)
JPY/EUR
5
15
30
-1.607
-0.895
-0.570
1.633
0.885
0.567
1.089
0.585
0.379
0.234
0.397
0.411
12.711
11.241
11.997
-0.007
-0.033
-0.008
(0.545)
(0.007)
(0.495)
0.039
0.048
0.034
(0.016)
(0.002)
(0.039)
NOK/EUR
5
15
30
-1.232
-0.697
-0.446
1.402
0.747
0.484
0.854
0.487
0.319
0.251
0.162
-0.036
9.228
9.704
12.685
0.035
0.005
-0.068
(0.036)
(0.761)
(0.000)
0.036
0.017
0.083
(0.209)
(0.958)
(0.000)
NZD/EUR
5
15
30
-1.695
-0.932
-0.582
1.699
0.904
0.571
1.170
0.610
0.383
0.349
-0.188
-0.806
15.685
9.959
17.827
-0.044
-0.059
-0.061
(0.006)
(0.000)
(0.000)
0.040
0.073
0.066
(0.104)
(0.000)
(0.000)
SEK/EUR
5
15
30
-1.365
-0.730
-0.503
1.389
0.778
0.484
0.885
0.488
0.321
-0.148
0.087
-0.092
8.334
8.384
8.763
0.046
0.017
-0.039
(0.007)
(0.314)
(0.017)
0.036
0.048
0.072
(0.221)
(0.035)
(0.000)
AUS/GBP
5
15
30
-1.683
-0.918
-0.581
1.821
0.929
0.591
1.230
0.639
0.420
-0.229
-0.211
-1.893
17.100
13.506
44.503
-0.110
-0.022
-0.097
(0.000)
(0.157)
(0.000)
0.047
0.017
0.043
(0.029)
(0.944)
(0.045)
CAD/GBP
5
15
30
-1.709
-0.931
-0.602
1.722
0.913
0.580
1.152
0.604
0.392
-0.064
0.080
-0.080
12.740
8.627
9.988
-0.085
0.010
-0.129
(0.000)
(0.540)
(0.000)
0.040
0.029
0.051
(0.084)
(0.375)
(0.010)
CHF/GBP
5
15
30
-1.388
-0.766
-0.479
1.390
0.726
0.464
0.943
0.520
0.342
0.051
0.226
-0.877
13.442
16.612
28.940
-0.037
0.037
-0.059
(0.003)
(0.003)
(0.000)
0.067
0.032
0.048
(0.000)
(0.074)
(0.001)
EUR/GBP
5
15
30
-1.165
-0.598
-0.401
1.162
0.629
0.418
0.764
0.421
0.282
-0.193
-0.147
0.324
9.183
15.589
21.871
-0.041
0.035
-0.053
(0.001)
(0.004)
(0.000)
0.054
0.019
0.068
(0.001)
(0.662)
(0.000)
JPY/GBP
5
15
30
-1.692
-0.913
-0.578
1.757
0.952
0.612
1.181
0.640
0.419
0.516
0.338
-0.038
14.281
16.520
23.768
-0.039
0.013
-0.048
(0.001)
(0.294)
(0.000)
0.045
0.048
0.053
(0.003)
(0.001)
(0.000)
NZD/GBP
5
15
30
-1.877
-1.032
-0.648
1.938
1.045
0.633
1.314
0.691
0.456
0.264
-0.605
-2.661
15.240
16.852
62.103
-0.053
0.022
-0.159
(0.001)
(0.178)
(0.000)
0.027
0.051
0.083
(0.491)
(0.014)
(0.000)
Notes: see below.
23
p-value
Table 5: Spot Rate Dynamics (cont.)
Spot Rate Changes (bps. per minute)
D:
Temporal Dependence
horizon
5%
95%
std
skew
kurtosis
Autocorrelation
p-value
Independence
p-value
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
(xi)
AUS/USD
5
15
30
-1.693
-0.905
-0.591
1.687
0.883
0.562
1.160
0.610
0.399
0.086
-0.088
0.411
18.120
12.623
13.210
-0.087
-0.030
-0.041
(0.000)
(0.015)
(0.001)
0.054
0.033
0.034
(0.000)
(0.075)
(0.044)
CAD/USD
5
15
30
-1.467
-0.776
-0.505
1.435
0.778
0.488
0.921
0.510
0.329
-0.085
0.290
-0.103
8.762
10.587
13.586
-0.003
-0.025
-0.044
(0.789)
(0.039)
(0.000)
0.023
0.053
0.043
(0.428)
(0.000)
(0.004)
DKK/USD
5
15
30
-1.578
-0.822
-0.567
1.548
0.831
0.549
1.014
0.536
0.351
0.095
0.094
-0.103
7.817
6.901
8.885
-0.015
0.012
0.022
(0.358)
(0.480)
(0.187)
0.050
0.048
0.048
(0.024)
(0.036)
(0.025)
NOK/USD
5
15
30
-2.089
-1.176
-0.730
2.184
1.184
0.784
1.352
0.747
0.490
0.094
0.168
-0.049
6.047
6.938
8.592
0.011
0.000
-0.048
(0.523)
(0.995)
(0.004)
0.032
0.031
0.031
(0.325)
(0.379)
(0.320)
SEK/USD
5
15
30
-2.304
-1.215
-0.810
2.276
1.204
0.784
1.436
0.783
0.511
-0.076
0.211
-0.057
6.168
8.700
8.471
0.012
0.012
-0.012
(0.477)
(0.487)
(0.468)
0.023
0.047
0.025
(0.710)
(0.039)
(0.587)
SGD/USD
5
15
30
-0.736
-0.434
-0.284
0.813
0.432
0.285
0.523
0.278
0.181
0.094
-0.046
0.128
9.615
9.321
8.823
-0.027
-0.036
-0.059
(0.121)
(0.033)
(0.000)
0.059
0.043
0.062
(0.016)
(0.105)
(0.003)
Notes: Columns (iii) - (vii) report statistics on the distribution of changes in the log spot rates over horizons h of 5, 15, and 30 minutes. The change
in rates are expressed in basis points per minutes, i.e., h st ⌘ (ln(St+h ) ln(St )) ⇤ 10000/h for h = {5, 15, 60}, where St is the mid-point rate at
time t. All statistics are computed from 10000 starting times t sampled at random from the span of the available time series for each currency pair.
Columns (viii) and (ix) report the first-order autocorrelation in h st (i.e. corr( h st+h , h st )) and the p-value for the null of a zero autocorrelation,
respectively. Column (x) reports the KS test for the null that the two conditional distributions f ( h st+h | h st > 0) and f ( h st+h | h st  0) are the
same. The asymptotic p-value for the null is shown in column (xi).
The temporal dependence of intraday rate changes documented in Table 5 might appear surprising to
someone familiar with the statistical properties of asset price changes measured over much longer horizons
(e.g., days, months or quarters). In particular, it would seem from the estimated autocorrelations that
future rate changes are (to some degree) forecastable using past rates; an apparent contradiction of Weakform efficiency. However, two caveats are in order. First, these correlations are computed from the midpoints of bid and ask rates. As such, the estimated autocorrelations do not imply that the future returns
available to traders (i.e. changes in log rates that account for the bid/o↵er spread) can be forecast. As
we shall see below, the forecastability of future forex returns adjusted for the spread is typically much
less than the apparent forecastability implied by the estimated autocorrelation in mid-point rate changes.
The second caveat concerns risk. Even in cases where there is forecastability for returns (adjusted for the
spread), the precision of the forecast is very low. Traders taking speculative positions based on the forecasts
would be exposed to significant risk of loss. Indeed, the risk of losses are so large relative to the expected
gains, trading strategies exploiting forecastability would look very unattractive when judged by standard
performance metrics like Sharpe ratios and Drawdown statistics. Section 7 examines the incentives facing
traders to exploit serial correlation in spot rate changes in greater detail.
The statistics in Table 5 are based on the entire span of the time series of intraday rates for each currency
24
pair. This span covers a decade for 14 pairs during which the structure of trading in the forex market changed
significantly. In addition, the data series span the 2008/9 world financial crisis. Consequently, it is possible
that the characteristics identified above mask secular changes in the behavior of rates as forex trading
institutions evolved and/or are unduly influenced by the atypical behavior of rates during the hight of the
financial crisis.
The statistics in Table 6 shed light on these issues. Columns (iii) - (vii) and (viii) - (xii) report statistics
on the distribution of rate changes (basis points per minute) between Jan 1st 2004 and Dec 31st. 2007, and
between Jan 1st. 2010 and Dec. 31st. 2013.16 Both of these subsamples cover periods that are far removed
from the hight of the 2008/9 crisis. To examine the stability of the rate-change distribution across the two
subsamples, I again use the KS test, and report its asymptotic p-value in the right-hand column of the table.
The statistics in Table 6 show that there has indeed been change in the rate-change distributions over
the past decade. Formally, this can be seen from the very small p-values for the KS tests reported in column
(xiv). A comparison of the statistics in columns (iii) - (vii) with those in (viii) - (xii) reveals that the tails of
the distributions, measured by the percentiles and kurtosis, generally exhibit the largest di↵erences across
the two subsamples. In other words, the incidence and size of atypical rate changes appears to have evolved
over the decade. That said, the majority of the statistics from the two subsamples are very similar. In
particular, the standard deviations are similar in size and decline with the rise in the horizon in the same
manner as their counterparts in Table 5. As above, this pattern is symptomatic of the generally negative
autocorrelation in rate changes that is present in both subsamples. Estimated autocorrelations (unreported)
are generally negative, and statistically significantly di↵erent from zero in the two subsamples, but the
estimates are uniformly small (in absolute value), like those in Table 6.
Figure 4 provides visual evidence that compliments the statistics reported in Tables 5 and 6. The figure
plots the rate-change densities for the four major currency pairs. Plot (i) in each panel shows density
functions for
h
st for h = {5, 15, 30} minutes in green, blue, and red, respectively. Here we can clearly see
how that dispersion of the densities increases as the horizon shortens from 30 to five minutes. Plot (ii) in
each panel shows the distributions from the pre-2008 and post-2009 subsamples. On close inspection it is
possible to see di↵erences between the densities, but they are extremely small. Moreover, the densities from
the subsamples do not look dissimilar to the densities in plot (i). Thus, while the di↵erences between the
subsample price-change distributions are statistically significant, the di↵erences in the estimated densities
do not appear economically important for the four major currency pairs. The Appendix shows that these
similarities carry over to the other currency pairs. Despite the large institutional changes in forex trading
over the past decade, the intraday dynamics of rates away from the Fix (and other scheduled announcements)
appears to have been stable.
16 I
only include statistics for currency pairs with reliable intraday data starting in 2004.
25
Table 6: Stability of Spot Rate Dynamics
2004-2007
A:
B:
C:
D:
2010-1013
std
skew
kurtosis
KS Test
p-value
(x)
(xi)
(xii)
(xiv)
1.462
0.737
0.487
0.890
0.463
0.299
0.193
0.135
0.267
6.333
6.094
5.886
0.000
0.001
0.000
-1.580
-0.805
-0.530
1.528
0.767
0.489
1.012
0.522
0.325
-0.652
-0.416
-0.559
14.041
9.485
9.684
0.000
0.001
0.201
8.780
7.115
9.113
-1.173
-0.603
-0.415
1.099
0.610
0.386
0.757
0.392
0.261
-0.174
0.568
0.337
9.364
9.330
8.359
0.000
0.001
0.021
0.506
-0.854
-1.450
18.506
35.074
49.030
-1.226
-0.677
-0.408
1.232
0.647
0.452
0.782
0.440
0.282
0.141
0.415
0.489
8.286
9.644
8.096
0.000
0.000
0.003
0.488
0.267
0.171
0.871
1.205
0.614
22.664
32.256
34.213
-1.059
-0.617
-0.371
1.082
0.542
0.371
0.764
0.405
0.253
0.014
0.101
0.433
31.187
22.086
17.937
0.000
0.000
0.000
1.331
0.728
0.728
1.000
0.532
0.352
0.562
0.679
0.562
23.980
17.937
20.992
-1.711
-0.943
-0.608
1.743
0.967
0.600
1.104
0.595
0.383
0.085
0.339
0.270
6.712
8.313
7.020
0.000
0.000
0.000
-1.146
-0.646
-0.646
1.216
0.612
0.612
0.815
0.459
0.308
0.447
0.909
-1.634
14.632
28.974
62.633
-1.496
-0.803
-0.482
1.392
0.738
0.502
0.987
0.532
0.334
-0.404
-0.001
-0.101
15.134
12.160
9.787
0.000
0.000
0.001
5
15
30
-0.895
-0.495
-0.495
0.903
0.503
0.503
0.667
0.365
0.244
-0.108
-0.516
0.968
12.598
25.800
46.873
-1.147
-0.613
-0.431
1.215
0.646
0.418
0.761
0.422
0.274
-0.185
-0.228
-0.210
7.534
11.588
7.546
0.000
0.000
0.000
JPY/GBP
5
15
30
-1.533
-0.814
-0.814
1.547
0.832
0.832
1.144
0.608
0.405
0.952
0.481
-0.597
22.271
27.077
41.379
-1.619
-0.880
-0.538
1.614
0.919
0.598
1.045
0.573
0.372
0.177
0.466
0.391
7.440
9.680
9.238
0.001
0.007
0.038
AUS/USD
5
15
30
-1.658
-0.897
-0.897
1.562
0.849
0.849
1.221
0.639
0.420
0.350
-0.161
0.418
24.528
16.309
16.003
-1.557
-0.827
-0.511
1.559
0.757
0.500
0.968
0.500
0.323
-0.052
0.162
0.365
7.257
6.025
7.591
0.000
0.002
0.001
CAD/USD
5
15
30
-1.496
-0.787
-0.787
1.467
0.787
0.787
0.946
0.528
0.342
0.024
0.575
0.020
10.811
12.847
16.472
-1.207
-0.700
-0.419
1.217
0.650
0.415
0.782
0.416
0.264
-0.211
-0.084
0.004
6.614
6.713
7.356
0.000
0.008
0.004
horizon
5%
95%
std
skew
kurtosis
5%
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
EUR/USD
5
15
30
-1.236
-0.671
-0.671
1.234
0.636
0.636
0.833
0.436
0.282
0.029
-0.466
-0.208
11.279
9.937
10.384
-1.369
-0.727
-0.478
CHF/USD
5
15
30
-1.324
-0.725
-0.725
1.404
0.753
0.753
0.889
0.474
0.294
0.436
0.268
0.018
9.868
7.398
6.575
JPY/USD
5
15
30
-1.235
-0.658
-0.658
1.312
0.673
0.673
0.829
0.428
0.272
0.151
0.007
-0.291
USD/GBP
5
15
30
-1.261
-0.648
-0.648
1.216
0.653
0.653
0.887
0.487
0.322
CHF/EUR
5
15
30
-0.644
-0.360
-0.360
0.695
0.382
0.382
JPY/EUR
5
15
30
-1.360
-0.742
-0.742
CHF/GBP
5
15
30
EUR/GBP
95%
(ix)
Notes: Columns (iii) - (vii) and (viii) - (xii) report statistics on the distribution of changes in the log quotes over horizons h of 5, 15,
and 30 minutes from quotes made between Jan 1st 2004 and Dec 31st. 2004, and between Jan 1st. 2010 and Dec. 31st. 2013. The
change in quotes are expressed in basis points per minutes, i.e., h st ⌘ (ln(St+h ) ln(St ))10000/h for h = {5, 15, 60}. All statistics
are computed from 10000 starting times t sampled at random. Column (xiv) reports the asymptotic p-value from the KS test of the
null that the distributions from the two subsamples are the same.
26
27
2
4
0
2
4
−2
0
2
4
4
4
4
2
2.5
0
−4
0.5
1
1.5
2
2.5
0
−4
0.5
1
1.5
2
0
−4
0
−4
0.5
−2
−2
−2
−2
0
iii: USD/GBP
0
i: USD/GBP
0
iii: CHF/USD
0
i: CHF/USD
2
2
2
2
4
4
4
4
0
−4
0.2
0.4
0.6
0.8
1
0
−4
0.5
1
1.5
2
2.5
0
−4
0.2
0.4
0.6
0.8
1
0
−4
0.5
1
1.5
2
2.5
−2
−2
−2
−2
0
iv: USD/GBP
0
ii: USD/GBP
0
iv: CHF/USD
0
ii: CHF/USD
2
2
2
2
4
4
4
4
Notes: Plots (i) shows the density functions for h st for h = {5, 15, 30} minutes in green, blue, and red, respectively. Plot (ii) shows the density functions h st from pre-2008 and
post 2009 data with solid and dotted lines, respectively. Plots (iii) and (iv) show the conditional densities for f ( h st | h st h > + ) (solid) and f ( h st | h st h <  ) (dotted),
where where + and  denote the upper and lower percentiles of the price-change distribution, respectively: equal to {75%, 25%} in plot (iii) and {97.5%, 2.5%} in plot (iv).
0
−4
0.5
1
0.5
iv: JPY/USD
2
2
2
1
1
1.5
0
ii: JPY/USD
0
iv: EUR/USD
0
0
−4
0.5
1
1.5
2
1.5
−2
iii: JPY/USD
−2
−2
−2
ii: EUR/USD
1.5
2
2.5
0
−4
0.5
0.5
4
1
0
−4
1.5
1
0
−4
1.5
2
4
2
0
2
2
i: JPY/USD
0
2.5
−2
−2
0.5
1
1.5
2.5
0
−4
0.5
1
1.5
2
2.5
3
iii: EUR/USD
0.5
0.5
0
1
1
−2
1.5
1.5
0
−4
2
2
0
−4
2.5
i: EUR/USD
2.5
Figure 4: Rate Change Densities
5
Pre-Fix Spot Rate Dynamics
In now turn to the central focus of this study; the behavior of spot rates in the periods immediately before
and after the 4:00 pm Fix. In this section I examine the pre-Fix behavior of rates between 3:00 and 4:00 pm
using the distributions for rate-changes away from the Fix as a benchmark to identify atypical behavior.
Figure 5 shows the rate-change densities over windows of {60,15,5,1} minutes before 4:00 pm for the
four major currency pairs. For each horizon and currency pair the figure plots the densities for spot rate
changes away from the Fix (discussed in Section 4) together with the densities for the pre-Fix rate changes
on intra-month and end-of-month days. The densities for the pre-Fix changes use the Fix as the end spot
rate in each rate change. For example, the density for end-of-month five-minute pre-Fix change is estimated
from the change in spot rates between 3:55 and 4:00 pm at the end of every month. The intra-month density
is similarly estimated from intraday data on all the other days. Notice, also, that these densities are for rate
changes expressed in basis points, rather than basis point per minute as in Figure 4.
Two features stand out from the plots in Figure 5. First, the behavior of pre-Fix rate changes are quite
unlike that of rate changes associated with normal trading activity. As the plots clearly show, the estimated
densities for the pre-Fix changes are quite di↵erent from the densities for rate-changes away from the Fix. It
appears that many pre-Fix rate changes are atypical of the changes we observe at other times. This visual
evidence is confirmed by KS tests for the equality of the pre-Fix and away-from-the-Fix distributions; they
give very small p-values for all currency pairs and horizons.
Second, the behavior of pre-Fix rate changes at the end of the month appear more atypical than those
on other days. Recall from Section 1 that there is a strong hedging incentive for fund managers and
derivative investors to submit fill-at-fix forex orders at the end of the month. The density plots show that
this institutional factor has a material a↵ect on the behavior of rates before the Fix. More specifically,
the dispersion of pre-Fix rate changes at the end of the month is significantly larger than the dispersion of
changes away from the Fix, and the dispersion of pre-Fix changes during the month. These di↵erences are
more pronounced at shorter horizons (particularly below 15 minutes). These density plots imply that the
Fix established at the end of each month is quite often far from the rates at which forex was trading less than
15 minutes earlier, and that rate changes (over the same horizon) of a similar size are extraordinarily rare in
trading away from the Fix. Importantly, this striking feature of the data applies to all 21 the currency pairs.
As the Appendix shows, the plots in Figure 5 are representative of the plots for the other currency pairs.
How atypical are the spot rate movements before the Fix? To answer this question, I compare the pre-Fix
rate changes to the tail probabilities from the distribution of rate-changes away from the Fix. Specifically, I
compute the fraction of days where the absolution pre-Fix change is larger than the 95th. percentile of the
distribution of absolute changes away from the Fix.17 If pre-Fix changes are consistent with normal trading
away from the Fix, they should be above the 95th. percentile on approximately one day in twenty (i.e., 5
percent of the time).
Table 7 reports the percentage of end-of-month and intra-month days on which the pre-Fix absolute basis
point change in spot rates is larger than the 95th. percentile threshold across horizons ranging from one to
60 minutes. The results in the table are quite remarkable. Notice, first, that the incidence of unusually large
17 The
distribution of absolute rate changes away from the Fix is estimated from the same random sample of 10000 rates for
each currency pair examined in Section 4.
28
29
10
20
50
100
10
20
−10
−10
0
JPY/USD 1 min
0
JPY/USD 15 mins
0
EUR/USD 1 min
0
10
10
EUR/USD 15 mins
20
50
20
50
0
−20
0.05
0.1
0.15
0.2
0
−100
0.01
0.02
0.03
0.04
0
−20
0.02
0.04
0.06
0.08
0.1
0.12
0
−100
0.01
0.02
0.03
0.04
0
CHF/USD 5 mins
0
10
50
−10
−50
0
USD/GBP 5 mins
0
10
50
USD/GBP 60 mins
−10
−50
CHF/USD 60 mins
20
100
20
100
0
−20
0.1
0.2
0.3
0.4
0
−50
0.02
0.04
0.06
0.08
0.1
0
−20
0.1
0.2
0.3
0.4
0
−50
0.02
0.04
0.06
0.08
Notes: Distribution for rate changes (in basis points) away from Fixes (black), intra-month pre-Fix (blue), and end-of-month pre-Fix (red).
0
−20
0.1
0.05
0
0.2
0.1
−10
0.3
0.15
0
−20
0.4
0.2
JPY/USD 5 mins
0.02
0.01
0
0.04
0.02
−50
0.06
0.03
0
−50
0.08
0.04
0
−100
0.1
JPY/USD 60 mins
0.05
0
−20
0.1
0.05
0
−20
0.2
0
−50
0.1
0
100
0.3
−10
50
0.15
EUR/USD 5 mins
0
0.02
0.04
0.06
0.08
0.1
0.4
−50
EUR/USD 60 mins
0.2
0
−100
0.01
0.02
0.03
0.04
Figure 5: Pre-Fix Rate Change Densities
0
10
−10
0
USD/GBP 1 min
0
10
USD/GBP 15 mins
−10
CHF/USD 1 min
0
CHF/USD 15 mins
20
50
20
50
30
CHF/EUR
JPY/EUR
NOK/EUR
NZD/EUR
SEK/EUR
Average
AUS/GBP
CAD/GBP
CHF/GBP
EUR/GBP
JPY/GBP
NZD/GBP
Average
AUS/USD
CAD/USD
DKK/USD
NOK/USD
SEK/USD
SGD/USD
Average
B:
C:
D:
22.414
23.276
13.559
8.065
13.559
8.197
14.845
18.841
19.718
17.241
21.739
18.103
17.910
18.926
18.966
17.094
16.129
26.471
16.949
19.122
28.448
31.897
15.254
22.581
20.339
11.475
21.666
30.435
28.169
30.172
31.304
27.586
25.373
28.840
23.276
28.205
29.032
30.882
25.424
27.364
22.222
21.698
28.846
27.586
25.088
(ii)
(i)
16.239
21.698
17.308
18.966
18.553
30
60
23.276
29.310
10.170
19.355
23.729
9.836
19.279
34.783
30.986
37.069
40.000
32.759
25.373
33.495
25.862
29.915
24.194
29.412
30.509
27.978
18.803
21.698
38.462
29.310
27.068
(iii)
15
29.310
30.172
11.864
25.807
23.729
14.754
22.606
34.783
29.578
37.069
41.739
34.483
23.881
33.589
28.448
34.188
35.484
36.765
38.983
34.774
14.530
20.755
42.308
35.345
28.234
(iv)
10
32.759
34.483
18.644
29.032
33.898
16.393
27.535
34.783
38.028
31.035
37.391
43.966
26.866
35.345
29.310
42.735
35.484
41.177
45.763
38.894
22.222
25.472
47.115
33.621
32.108
(v)
5
46.552
43.966
30.509
46.774
40.678
19.672
38.025
56.522
39.437
50.000
50.435
56.035
47.761
50.031
33.621
52.137
58.065
48.529
45.763
47.623
33.333
37.736
61.539
51.724
46.083
(vi)
1
10.092
11.273
10.575
9.724
9.792
7.833
9.881
8.537
9.811
7.158
6.926
7.775
9.865
8.345
7.375
9.418
10.070
12.306
9.472
9.728
10.248
9.832
10.027
7.394
9.375
(i)
60
12.427
16.722
10.881
12.481
12.336
9.667
12.419
12.940
14.614
10.923
10.603
10.132
13.272
12.081
9.987
10.574
14.330
16.549
13.975
13.083
11.653
13.242
12.114
10.822
11.958
(ii)
30
11.259
15.183
6.820
9.954
11.334
8.917
10.578
13.008
16.238
11.378
12.399
10.008
12.420
12.575
9.819
8.013
14.562
15.559
16.149
12.820
9.380
10.939
11.071
9.665
10.264
(iii)
15
11.426
14.642
7.433
11.792
10.948
9.500
10.957
13.415
16.847
12.371
11.372
10.091
14.195
13.048
11.125
8.550
14.795
15.842
15.761
13.215
7.521
10.895
10.481
9.748
9.661
(iv)
10
II: Intra-Month
13.136
16.889
7.126
12.864
11.411
10.083
11.918
14.160
22.463
12.743
12.185
11.373
21.221
15.691
11.589
10.905
19.597
20.368
15.450
15.582
7.107
9.433
10.799
11.276
9.654
(v)
5
19.516
26.040
10.575
24.043
22.282
18.667
20.187
26.423
30.176
21.804
22.488
21.464
30.518
25.479
15.086
15.572
29.202
27.581
29.115
23.311
10.496
14.969
22.051
20.446
16.990
(vi)
1
II the percentage for intra-month rate changes. Averages for the currencies in each block are reported in the last row.
Notes: Each cell reports the percentage of days in which the absolute basis point change in rates in the window before the Fix is larger than the 95th.
percentile from the distribution of absolute basis point rate changes away from the Fix. Panel I reports the percentage for end-of-month rate changes, panel
EUR/USD
CHF/USD
JPY/USD
USD/GBP
Average
A:
horizon
I: End-of-Month
Table 7: Tail Probabilities for pre-Fix Rate Changes
pre-Fix rate changes is much higher at the end of the month than on other days. This pattern holds across
all the currency pairs and over all the horizons. It reinforces the visual evidence in Figure 5 indicating that
pre-Fix spot rate dynamics at the end of the month are di↵erent from other days. Second, the incidence of
unusually large pre-Fix changes rises as the horizon shortens. This means that if we compare the level of
the Fix with the level of rates in the prior hour on a randomly chosen day, we are likely to see an unusually
large jump in rates shortly before 4:00 pm.
Perhaps the single most striking aspect of Table 7 concerns the high incidence of unusually large rate
movements immediately prior to Fix. Examples of large price movements immediately before 4:00 pm on
particular days for specific currencies have been reported in the financial press (see, e.g., Reuters 2013). The
statistics in Table 7 show that unusually large pre-Fix rate changes are almost commonplace. For example,
atypically large changes in the minute before the Fix on intra-month days occur at more than three times
the rate that would be consistent with normal trading activity across the four major currency pairs, and
at higher rates across the other currency pairs. The incidence of atypically large rate changes immediately
before the Fix is even higher at the end of the month. At the one minute horizon atypical changes occur
between four and twelve times the rate consistent with normal trading activity. These are remarkably high
numbers. For two of the major currency pairs, the JPY/USD and USD/GBP, atypically large rate changes
in the minute before 4:00 pm occur at more than ten times the rate consistent with normal trading activity.
It is also informative to examine the incidence of atypically large pre-Fix rate changes through time. For
this purpose Table 8 reports the number of atypical changes (again using the 95th. percentile threshold) over
a one minute horizon at the end of the month during each year covered by the dataset. P-values for the null
hypothesis that the number of atypical end-of-month changes occurs by chance (based on the distribution
of absolute rate changes in normal forex trading) are reported in parenthesis. As the table clearly shows,
the high incidence of atypically large pre-Fix rate changes is not concentrated in a few years or currency
pairs. On the contrary, it is pervasive. For example, in the case of the USD/GBP, there have been a high
number of atypically large changes in every year between 2004 and 2013. In fact the numbers are so high
in nine of the years that the probability of this representing rate movements from normal forex trading in
USD/GBP in any year is less than 0.001 (i.e., less that one in one thousand). This repeated high incidence
of atypically large pre-Fix rate changes is also evident in the JPY/USD, JPY/EUR, CHF/GBP, EUR/GBP,
JPY/GBP,USD/USD and CAD/USD. The results in Table 8 also show that the peak incidence of atypically
large rate changes did not occur around the world financial crisis. Aggregating across all 21 currency pairs,
the peak year was 2010 with a total of 148.
31
Table 8: Pre-Fix Tail Events By Year (1 minute window)
A:
EUR/USD
CHF/USD
JPY/USD
USD/GBP
B:
CHF/EUR
JPY/EUR
2004
2005
2006
2007
2008
2009
2010
2011
2012
2
(0.165)
1
(0.450)
3
(0.011)
6
(0.000)
5
(0.000)
4
(0.001)
4
(0.001)
5
(0.000)
1
(0.600)
0
(0.569)
7
(0.000)
6
(0.000)
6
(0.000)
5
(0.000)
11
(0.000)
3
(0.028)
5
(0.000)
3
(0.007)
5
(0.000)
5
(0.000)
6
(0.000)
4
(0.002)
6
(0.000)
9
(0.000)
6
(0.000)
5
(0.000)
9
(0.000)
7
(0.000)
3
(0.028)
7
(0.000)
8
(0.000)
8
(0.000)
4
(0.003)
7
(0.000)
4
(0.003)
5
(0.000)
2
(0.138)
4
(0.002)
7
(0.000)
7
(0.000)
4
(0.003)
6
(0.000)
1
(0.550)
4
(0.002)
1
(0.550)
4
(0.002)
3
(0.028)
7
(0.000)
4
(0.003)
8
(0.000)
1
(0.200)
8
(0.000)
1
(0.200)
4
(0.003)
8
(0.000)
8
(0.000)
7
(0.000)
4
(0.003)
9
(0.000)
9
(0.000)
8
(0.000)
5
(0.000)
7
(0.000)
7
(0.000)
5
(0.000)
10
(0.000)
4
(0.003)
6
(0.000)
0
(0.540)
5
(0.000)
6
(0.000)
5
(0.000)
5
(0.000)
6
(0.000)
5
(0.000)
4
(0.002)
4
(0.002)
6
(0.000)
10
(0.000)
6
(0.000)
7
(0.000)
7
(0.000)
7
(0.000)
6
(0.000)
9
(0.000)
5
(0.000)
7
(0.000)
8
(0.000)
9
(0.000)
8
(0.000)
8
(0.000)
6
(0.000)
8
(0.000)
9
(0.000)
10
(0.000)
7
(0.000)
6
(0.000)
4
(0.003)
7
(0.000)
7
(0.000)
6
(0.000)
6
(0.000)
5
(0.000)
4
(0.003)
7
(0.000)
8
(0.000)
6
(0.000)
4
(0.003)
2
(0.138)
3
(0.021)
7
(0.000)
6
(0.000)
9
(0.000)
2
(0.138)
9
(0.000)
6
(0.000)
3
(0.001)
2
(0.015)
3
(0.001)
2
(0.015)
9
(0.000)
5
(0.000)
5
(0.000)
8
(0.000)
3
(0.021)
3
(0.028)
9
(0.000)
4
(0.003)
6
(0.000)
6
(0.000)
7
(0.000)
3
(0.028)
4
(0.003)
3
(0.028)
2
(0.165)
6
(0.000)
5
(0.000)
1
(0.600)
3
(0.028)
7
(0.000)
1
(0.600)
4
(0.003)
3
(0.028)
1
(0.600)
4
(0.002)
8
(0.000)
2
(0.138)
4
(0.002)
5
(0.000)
2
(0.138)
NOK/EUR
NZD/EUR
SEK/EUR
C:
AUS/GBP
CAD/GBP
CHF/GBP
EUR/GBP
JPY/GBP
4
(0.003)
3
(0.028)
4
(0.003)
3
(0.021)
3
(0.021)
3
(0.021)
4
(0.002)
4
(0.002)
4
(0.002)
5
(0.000)
4
(0.003)
8
(0.000)
4
(0.002)
4
(0.003)
3
(0.021)
3
(0.021)
5
(0.000)
5
(0.000)
4
(0.003)
7
(0.000)
NZD/GBP
D:
AUS/USD
CAD/USD
DKK/USD
NOK/USD
SEK/USD
SGD/USD
2013
Notes: Each cell reports the number of months in each year that where the absolute change in rates in the 1 minute before
the Fix falls in the 95th percentile of the empirical distribution of rate changes away from the Fix. P-values for the null that
the number of months occurs by purely by chance are reported in parentheses.
To summarize, the results above show that the changes in forex rates observed immediately before the
4:00 pm Fix are extraordinarily unusual when compared to their behavior in normal trading away from the
32
Fix: rates regularly jump by an amount that is very rarely seen elsewhere. Moreover, the incidence of these
atypically large pre-Fix rate changes is particularly high at the end of each month, appears pervasive across
currency pairs and through time.
6
Post-Fix Spot Rate Dynamics
The high incidence of unusually large changes in spot rates immediately before Fix carries over into the
behavior of rates after 4:00 pm. Table 9 reports the incidence of large post-Fix rate changes (starting at the
Fix) over horizons of one to 60 minutes. As above I use the 95th. percentile threshold from the empirical
distribution of absolute price changes away from the Fix to identify atypically large rate changes, and report
their incidence for each of the exchange rate pairs at the end of each month and on other intra-month days.
The results in Table 7 show that the incidence of atypically large post-Fix rate changes di↵ers from the
incidence of the pre-Fix counterparts. For example, the statistics in Panel II show the incidence of unusually
large rate movements falls as the horizon lengthens. At the one and five minute horizons, the incidence is
approximately twice as high as we would expect to see in trading away from the Fix, but atypically large
rate changes over 60 minutes occur at close to the normal frequency. By this metric, most of the unusual
behavior in rates on intra-month days is confined to the first few minutes following 4:00 pm. In contrast,
Table 7 showed that unusual rate behavior is evident up to 30 minutes before the Fix on intra-month days.
The behavior of the spot rates at the end of the month is distinctly di↵erent. As panel I of Table 9 shows,
the incidence of atypically large rate changes is larger at all horizons. For most currency pairs, the incidence
at the one minute horizon is at least four times higher than we would expect to see in normal trading,
declining to between two and three times normal at the 30 minute horizon. While high, these incidence rates
are well below those reported in Table 7 for pre-Fix changes over comparable horizons.
Together, the statistics in Tables7 and 9 clearly establish that rates are unusually volatile immediately
before and after the Fix, particularly at the end of the month. I now consider how the pre- and post-Fix behavior of rates are linked. For this purpose I estimate the bivariate density for pre- and post-Fix rate changes
at di↵erent horizons. More specifically, I estimate the bivariate density g(ln(St+h /Stf ix ), ln(S ft ix /St
h )).
In
view of the results above, I focus on the behavior of rates at the end of each month, and so use the rates from
those days to estimate the bi-variate density g(., .). Estimation uses a Gaussian Kernel with the bandwidth
determined as in Bowman and Azzalini (1997).
Figure 6 shows the density functions for the four major currency pairs at horizons ranging from 15 to
one minute. (Plots for the 17 other currency pairs are in the Appendix.) Each plot shows the contours of
the estimated density, g(., .), where the pre- and post-Fix rate changes are expressed in basis points. Notice
that the horizontal (pre-Fix) and vertical (post-Fix) axes have the same scale in each plot, but di↵er across
plots. Each plot also shows a solid line that represents the projection (i.e. regression) of ln(St+h /Stf ix ) on
ln(S ft ix /St
h ),
denoted as P(ln(S ft ix /St
h )).
This line provides information on the intertemporal dependence
between the pre- and post-Fix rate changes discussed below.
The plots in Figure 6 contain a lot of information about the behavior of spot rates immediately before
and after the Fix. Consider, first, the general shape of the density contours. In all cases, the maximum
width of each contour exceeds its maximum hight. This feature is present in the bivariate densities across
33
34
CHF/EUR
JPY/EUR
NOK/EUR
NZD/EUR
SEK/EUR
Average
AUS/GBP
CAD/GBP
CHF/GBP
EUR/GBP
JPY/GBP
NZD/GBP
Average
AUS/USD
CAD/USD
DKK/USD
NOK/USD
SEK/USD
SGD/USD
Average
B:
C:
D:
7.759
8.621
5.085
4.839
11.864
8.197
7.727
7.246
11.268
6.897
5.217
3.448
11.940
7.669
6.897
4.274
8.065
10.294
10.170
7.940
10.345
18.103
11.864
14.516
16.949
4.918
12.783
20.290
19.718
11.207
14.783
11.207
20.896
16.350
10.345
12.821
8.065
22.059
11.864
13.031
14.530
15.094
18.269
15.517
15.853
(ii)
(i)
5.983
6.604
4.808
5.172
5.642
30
60
18.103
17.241
10.170
14.516
11.864
3.279
12.529
23.188
19.718
14.655
19.130
15.517
13.433
17.607
16.379
16.239
4.839
16.177
13.559
13.439
15.385
18.868
16.346
14.655
16.313
(iii)
15
17.241
16.379
11.864
19.355
10.170
6.557
13.594
20.290
19.718
17.241
18.261
13.793
11.940
16.874
14.655
14.530
8.065
19.118
13.559
13.985
11.966
17.925
20.192
14.655
16.184
(iv)
10
27.586
30.172
15.254
22.581
18.644
8.197
20.406
28.986
33.803
20.690
19.130
15.517
32.836
25.160
19.828
18.803
12.903
26.471
16.949
18.991
17.094
18.868
21.154
13.793
17.727
(v)
5
24.138
30.172
18.644
24.194
28.814
27.869
25.638
26.087
23.944
21.552
26.087
14.655
31.343
23.945
16.379
25.641
20.968
41.177
40.678
28.969
20.513
26.415
21.154
18.103
21.546
(vi)
1
5.797
5.990
5.287
4.211
4.780
4.667
5.122
5.488
5.345
3.889
3.292
4.839
5.820
4.779
5.310
5.370
3.408
5.375
3.882
4.669
4.917
4.827
4.492
3.965
4.550
(i)
60
9.425
9.942
8.736
8.959
8.790
7.167
8.836
7.859
7.375
7.199
6.156
7.568
7.239
7.233
8.681
8.468
7.591
8.911
7.531
8.236
9.711
9.965
8.439
8.137
9.063
(ii)
30
8.674
9.318
9.042
8.499
8.867
7.917
8.719
6.911
7.510
7.613
6.841
8.189
6.529
7.265
7.965
7.600
6.739
7.638
7.609
7.510
9.298
9.699
8.893
7.228
8.779
(iii)
15
7.923
9.235
9.349
9.495
8.867
8.083
8.825
7.656
6.698
7.737
7.054
6.989
7.452
7.264
8.807
8.674
7.436
7.992
7.531
8.088
8.554
8.946
9.392
7.683
8.644
(iv)
10
9.008
10.399
8.429
8.116
7.941
7.667
8.593
7.114
8.660
8.440
7.738
8.519
9.226
8.283
8.681
8.798
10.380
10.113
8.385
9.271
8.554
8.193
8.394
8.922
8.516
(v)
5
II: Intra-Month
7.381
9.193
5.900
11.792
11.103
14.667
10.006
8.537
8.187
9.102
10.389
7.610
11.001
9.137
8.260
7.435
18.048
11.245
16.537
12.305
6.157
6.997
9.483
6.939
7.394
(vi)
1
percentage for intra-month rate changes. Averages for the currencies in each block are reported in the last row.
Notes: Each cell reports the percentage of days in which the absolute basis point change in rates in the window after the Fix is larger than the 95 percentile
from the distribution of absolute basis point rate changes away from the Fix. Panel I reports the percentage for end-of-month rate changes, panel II the
EUR/USD
CHF/USD
JPY/USD
USD/GBP
Average
A:
horizon
I: End-of-Month
Table 9: Tail Probabilities for Post-Fix Rate Changes
post
post
post
20
40
0
5
10
−20
−20
20
40
0
5
10
−20
−20
−10
−10
10
20
−20
−20
0
pre
JPY/USD 5 mins
0
pre
−10
0
10
20
−5
−10
−20
JPY/USD 15 mins
−10
0
10
20
−40
−40
−20
0
20
40
−10
−10
10
20
−20
−20
0
pre
EUR/USD 5 mins
0
pre
−10
0
10
20
−5
−10
−20
EUR/USD 15 mins
−10
0
10
20
−40
−40
−20
0
20
40
−5
−10
−5
−10
0
pre
JPY/USD 1 mins
0
pre
JPY/USD 10 mins
0
pre
EUR/USD 1 mins
0
pre
5
10
5
10
EUR/USD 10 mins
10
20
10
20
−20
−20
−10
0
10
20
−40
−40
−20
0
20
40
−20
−20
−10
0
10
20
−40
−40
−20
0
20
40
0
pre
CHF/USD 5 mins
0
pre
10
20
−10
−20
0
pre
USD/GBP 5 mins
0
pre
10
20
USD/GBP 15 mins
−10
−20
CHF/USD 15 mins
Figure 6: Bivariate Pre- and Post- Fix Rate Change Densities
20
40
20
40
−10
−10
−5
0
5
10
−20
−20
−10
0
10
20
−10
−10
−5
0
5
10
−20
−20
−10
0
10
20
−5
0
pre
CHF/USD 1 mins
0
pre
0
pre
USD/GBP 1 mins
0
pre
5
10
5
10
USD/GBP 10 mins
−10
−5
−10
CHF/USD 10 mins
10
20
10
20
Notes: Each plot shows the contours of the estimated bivariate density for pre- and post-fix rate changes (in basis points) over horizons of 1 to 15 minutes. The solid line in each
plot is the estimated regression line from the regression on the post-Fix rate change in the pre-Fix change. All estimates are based on end-of-month data.
post
post
post
post
post
post
post
post
post
post
post
post
post
35
all the currency pairs and at all horizons. Thus, rates are more volatile immediately before than after the
Fix. The plots in Figure 6 also show that there is no simple monotonic relation between the horizon and
the dispersion of the rate changes. While the dispersion at the one minute horizon is smaller than at the
15 minute horizon, for some currency pairs the pre- and post-Fix dispersions are larger at the five than ten
minute horizons, (see, e.g. CHF/USD and JPY/USD). This pattern is noteworthy because there would be
a monotonic relation between the (pre and post-Fix) dispersion and the horizon if log spot rates followed a
martingale.
The most significant information conveyed by the plots in Figure 6 concerns the temporal dependence
between the pre- and post-Fix rate changes. If post-Fix changes were distributed independently of the preFix change, the contour plots would be symmetric around the horizon dashed line. This is clearly not the
case for the four major currency pairs shown in Figure 6, nor is it so for any of the other 17 currency pairs.
Although the details di↵er by currency pair and horizon, in general the contours appear as ellipses that are
rotated clockwise around the point (0,0) (see, e.g., the contours for the USD/GBP at the ten-minute horizon).
This pattern implies that positive post-Fix price changes are more likely than negative changes if they were
preceded by a negative pre-Fix change, and vise-versa. Or, in terms of levels, if rates jumped up immediately
before the Fix, they are more likely to jump downwards immediately afterwards than upwards. Similarly,
rates are more likely to rise rather than fall immediately after 4:00 pm if they had fallen immediately before
the Fix. In sum, therefore, the densities show that there is a tendency for rates to revert back towards their
pre-Fix level immediately after 4:00 pm.
We can gauge the degree of rate reversion following the fix from the projection lines shown on each contour
plot. By definition the projection allows us to spilt the post-Fix price change, ln(St+h /Stf ix ), into a portion
that is perfectly correlated with the pre-Fix change, the projection P(ln(Stf ix /St
h ));
and a projection error,
⌘t+h , that is uncorrelated with the pre-Fix change:
ln(St+h /Stf ix ) = P(ln(Stf ix /St
The plots identify P(ln(Stf ix /St
h ))
h ))
+ ⌘t+h .
by the solid straight line. The vertical distances between the line and
the contours represent the dispersion in ⌘t+h conditioned on a particular pre-fix price change ln(Stf ix /St
h ).
As Figure 6 clearly shows, the projection lines slope downwards (from left to right) at all horizons and across
all four currency pairs. This pattern that is repeated across all the other 17 currency pairs. The steepness of
these slopes identifies the degree to which pre-Fix changes in the level of rates are reversed following the Fix.
For example, in the case of the USD/GBP, the projection line has a slope of approximately -0.4. This means
that a 10 basis point fall in the USD/GBP rate in the five minutes before the fix is, on average, followed by
a 4 basis point rise in the USD/GBP rate in the five minutes following the fix.
Table 10 provides more information on the projections across all 21 currency pairs. The table reports the
estimated projection coefficients, their (heteroskedastic-consistent) standard errors, and the uncentered R2
statistics for the projections over the horizons of {1, 5, 10, and 15} minutes. The estimated coefficients are
uniformly negative, ranging in value from -0.08 to -0.61. More than half are statistically significant at the
five percent level. The R2 statistics measure the variance contribution of the projections to the post-Fix rate
⌘
⇣
⌘
⇣
changes, V ar P(ln(Stf ix /St h )) /V ar ln(St+h /Stf ix ) . As the table shows, these statistics are generally
36
37
CHF/EUR
JPY/EUR
NOK/EUR
NZD/EUR
SEK/EUR
AUD/GBP
CAD/GBP
CHF/GBP
EUR/GBP
JPY/GBP
NZD/GBP
AUD/USD
CAD/USD
DKK/USD
NOK/USD
SEK/USD
SGD/USD
B:
C:
D:
(0.077)
(0.150)
(0.090)
(0.118)
(0.078)
(0.154)
(0.073)
(0.077)
(0.061)
(0.042)
(0.130)
(0.108)
(0.097)
(0.145)
(0.049)
(0.056)
(0.074)
(0.108)
(0.085)
(0.102)
(0.238)
-0.129
-0.107
-0.081
-0.201
-0.235⇤
-0.375⇤
-0.167⇤
-0.309⇤
-0.233⇤
-0.303⇤
-0.038
-0.267⇤
-0.228⇤
-0.147
-0.397⇤
-0.247⇤
-0.189⇤
-0.259⇤
-0.135
-0.237⇤
-0.443
Std Error
0.170
0.069
0.054
0.029
0.111
0.212
0.377
0.002
0.161
0.134
0.066
0.536
0.113
0.257
0.089
0.307
0.209
0.018
0.009
0.011
0.115
R2
-0.279⇤
-0.196⇤
-0.248
-0.203⇤
-0.203⇤
-0.142⇤
-0.324⇤
-0.039
-0.290⇤
-0.288⇤
-0.164
-0.413⇤
-0.257⇤
-0.386⇤
-0.232⇤
-0.339⇤
-0.280⇤
-0.092
-0.220
-0.090
-0.172
Coe↵
(0.068)
(0.080)
(0.138)
(0.090)
(0.104)
(0.211)
(0.037)
(0.115)
(0.087)
(0.106)
(0.133)
(0.041)
(0.078)
(0.159)
(0.054)
(0.068)
(0.084)
(0.094)
(0.172)
(0.064)
(0.123)
Std Error
10 Minutes
0.190
0.084
0.051
0.057
0.063
0.023
0.381
0.002
0.198
0.202
0.093
0.560
0.140
0.315
0.207
0.381
0.218
0.008
0.039
0.018
0.090
R2
-0.256⇤
-0.315⇤
-0.312
-0.169
-0.396⇤
-0.313
-0.431⇤
-0.344
-0.410⇤
-0.473⇤
-0.256
-0.505⇤
-0.199
-0.467⇤
-0.211⇤
-0.439⇤
-0.410⇤
-0.251
-0.112
-0.126
-0.357
Coe↵
(0.106)
(0.052)
(0.255)
(0.089)
(0.159)
(0.161)
(0.050)
(0.260)
(0.180)
(0.185)
(0.223)
(0.053)
(0.107)
(0.168)
(0.049)
(0.126)
(0.107)
(0.165)
(0.209)
(0.068)
(0.255)
Std Error
5 Minutes
0.144
0.140
0.079
0.043
0.161
0.156
0.464
0.079
0.298
0.365
0.149
0.633
0.104
0.408
0.162
0.447
0.307
0.060
0.015
0.051
0.243
R2
-0.124
-0.178⇤
-0.164
-0.079
-0.234⇤
-0.154
-0.031
-0.040
-0.150
-0.209⇤
-0.155⇤
-0.246⇤
-0.096
-0.605⇤
-0.075
-0.141
-0.199⇤
-0.150
-0.160
-0.164⇤
-0.105⇤
Coe↵
(0.080)
(0.064)
(0.102)
(0.086)
(0.068)
(0.309)
(0.050)
(0.103)
(0.085)
(0.047)
(0.039)
(0.075)
(0.129)
(0.200)
(0.110)
(0.118)
(0.070)
(0.082)
(0.138)
(0.045)
(0.046)
Std Error
1 Minute
0.061
0.071
0.065
0.014
0.126
0.015
0.008
0.003
0.079
0.168
0.179
0.239
0.020
0.633
0.009
0.061
0.068
0.048
0.035
0.173
0.066
R2
statistic of the null that the post-Fix rate change distributions conditioned on the sign of the pre-Fix change are equal.
Notes: The table reports the estimated projection coefficient, its (heteroskedastic consistent) standard error, and the R2 statistic from the projection of the post-fix rate change on the pre-fix
change over the horizons shown at the top of each panel. The “⇤ ” indicates statistical significance at the 5 percent level. The right hand column of each panel reports the p-value for the KS
EUR/USD
CHF/USD
JPY/USD
USD/GBP
A:
Coe↵
15 Minutes
Table 10: Post-Fix Projection Estimates
small (i.e. below 0.2). This indicates that most of the variation in post-Fix changes over time is attributable
to projection errors that are uncorrelated with the pre-Fix changes. Notable exceptions to this pattern
include the NZD/GBP, AUD/GBP, NZD/EUR and JPY/EUR rates. The R2 statistics are good deal larger
in these currency pairs; as high as 0.6 in the case of the NZD/GBP at the five-minute horizon. In these
cases, rate reversion accounts for a significant fraction of the time series variation in post-Fix rate changes.
The projection coefficients shown in Table 10 provide one set of estimates for the average degree of rate
revision following the Fix. By construction, these estimates assume that the rate revision is proportional to
the pre-Fix rate change, and does not depend on whether rates rose or fell towards the Fix. Alternatively, we
can estimate the size of spot rate revisions from the average path of rates after the Fix that are conditioned
on the pre-Fix changes. For example, we can examine the average paths for spot rates conditioned on pre-Fix
changes above or below certain thresholds. One advantage of this approach is that it can identify how the
degree of rate revision varies as we move further beyond the Fix.
Figure 7 plots the average spot rate paths in the two hours around the 4:00 pm for the four major currency
pairs. All the paths plotted in the figure are measured in basis points relative to the rate a 3:45 pm. The
horizontal axis shows minutes after the Fix; so -15 corresponds to 3:45 pm and 0 corresponds to 4:00 pm
(identified by the vertical line). Each plot shows six average spot rate paths that are conditioned on the
change in rates between 3:45 and 4:00 pm. I condition on the pre-Fix changes at this horizon because 3:45
pm is the cut-o↵ time for dealer-banks to accept fill-at-fix orders. The solid black line in each plot depicts
the average rate path across all end-of-month trading days where the pre-Fix price change is positive. The
dashed line depicts the analogous path when the pre-Fix change is negative. Average rate paths for intramonth days are shown by two dotted blue lines (the upper and lower lines are conditioned on positive and
negative pre-fix price changes, respectively). The remaining upper and lower lines (drawn with dashes and
dots) identify the average price paths on end-of-the month trading days where the pre-fix price change is in
the 75th. and 25th. percentiles of the pre-fix price change distribution, respectively. For the sake of clarity,
both the dotted and dash-dotted lines are hidden to the left of -15. As above, analogous plots for the other
17 currency pairs are in the Appendix.
The plots in Figure 7 provide a good deal of information about both the size and timing of the rate
revisions following the Fix. Consider, first, the paths on intra-month days (shown by the blue dotted lines).
These paths identify very small reversals during the first minute after the Fix (approximately equal to one
basis point). Thereafter the paths a flat. These patterns are common across all the currency pairs. They
are consistent with the idea that a new “equilibrium” rate is established based on the information contained
in Fix-related trading almost immediately after 4:00 pm. This doesn’t mean that rates remain at this level
on any particular day, they do not. Rather it implies that all the relevant information contained in trading
at (or immediately before) the Fix is fully assimilated into rates by approximately 4:01 pm so there is no
systematic tendency for rates to rise or fall after that.
The rate paths from end-of-month trading days are quite di↵erent. Consistent with the statistics on
pre-Fix rate volatility, changes in rates between 3:45 and 4:00 pm are larger (in absolute value). The plots
also show that generally it takes longer for the new post-Fix equilibrium rate to be established, and that it
tends to be further away from the extremum of the rate path. The di↵erences between the end-of-month
and intra-month paths is particularly clear cut in the case of the USD/GBP. Here the lowest average rate
38
Figure 7: Average Rate Paths Around the Fix
EUR/USD
CHF/USD
20
15
15
10
10
5
5
0
0
−5
−5
−10
−10
−15
−15
−60
−20
−45
−30
−15
0
15
30
45
60
−60
−45
−30
−15
39
JPY/USD
0
15
30
45
60
15
30
45
60
USD/GBP
25
20
20
15
15
10
10
5
5
0
0
−5
−5
−10
−10
−15
−15
−20
−20
−60
−45
−30
−15
0
15
30
45
60
−25
−60
−45
−30
−15
0
Notes: Average price path in basis points around 3:45 pm level conditioned on: (i) positive pre-fix changes (over 15 mins) at end of month
(solid black); (ii) negative pre-fix changes (over 15 mins) at end of month (dashed black); (iii) pre-fix changes above the 75th. percentile of
end-of-month distribution (upper red dashed dot); (iv) pre-fix changes in the 25th. percentile of end-of-month distribution (lower red dashed
dot); (v) positive and negative pre-fix changes on intra-month days (upper and lower blue dots).
(across all days when prices fell towards the Fix) is 15 basis points below its level at 3:45 pm. Thereafter,
rates immediately rebound by five basis points, before more falling back more slowly to produce a long-term
reversal of approximately two basis points. On days when rates rise towards the Fix, the average increase is
15 basis points. Rates then fall back until 4:15 for a total long-term reversal of 5 basis points.
The plots in Figure 7 also show average rate paths following unusually large pre-Fix rate changes (i.e.
those in the 75th. and 25th. percentiles of the empirical distribution) at the end-of-month trading days by
the dashed-dotted lines. In some cases these paths identify larger rate revisions than occur on average across
all end-of-month trading days, but in others the paths appear very similar. For example, in the case of the
EUR/USD there is approximately five basis point revision following unusually large rises in rates towards
the Fix, verses a revision of roughly one basis point on average across all end-of-month days. On the other
hand, the paths for the USD/GBP show little di↵erence in the size of the rate revisions following unusually
large pre-Fix changes and other end-of-month trading days.
One final feature of Figure 7 deserves particular comment. The paths in all the plots are conditioned on
the change in rates between 3:45 and 4:00 pm without regard to when rates changed within the 15-minute
window. Thus, if most of the movement in rates occurred immediately before the Fix, say between 3:59 and
4:00 pm, the paths would be flat until a point just to the left of the vertical line. Instead, the paths in Figure
7 show that on average rates start “drifting” upwards or downwards soon after 3:45 pm. In other words,
rates appear to “anticipate” whether the Fix will be above or below its level at 3:45 pm, and begin to move
in that direction well before 4:00 pm. This form of “anticipatory” rate behavior is not seen at other times
in the trading day.
7
Forex Trading Around the Fix
The behavior of forex rates around the 4:00 Fix is extremely unusual. When judged against the distribution
of rate dynamics away from the Fix, both the volatility and serial dependence of pre- and post-Fix rate
changes at the end-of-end month are quite extraordinary. This section provides an economic perspective on
these statistical findings. In particular I examine whether the behavior of rates could be consistent with the
e↵ective and efficient intermediation of forex orders around the Fix.
At face value many of the results in Section 6 appear inconsistency with Weak-form efficiency, a basic
measure of a well-functioning competitive market. In particular, the projection results in Table 10 and
the rate paths in Figure 7 suggest that information contained in pre-Fix rates can be used to forecast rate
movements after the Fix. More specifically, the projection coefficient estimates imply that, on average, endof-month rates fall after the Fix if they rose beforehand; or conversely, rates rise after the Fix if they fell
beforehand. Of course this forecasting pattern lies behind the average price paths in Figure 7. It suggests
the simple end-of-month trading strategy of taking a long (short) position at 4:00 pm if rates fell (rose)
towards the Fix. This strategy should generate positive returns on average, but actual returns on any day
could be positive or negative depending on the gap between the Fix and the rate obtained when the position
is closed. The question is: Would a trading strategy that exploits the forecastability of rates around the Fix
be attractive to market participants?
To address this question, I computed the realized returns on trading strategies that initiated long and
40
short positions at the end-of-month Fix with durations of h = {1, 5, 15} minutes. The long and short
positions are selected according to the change in rates over the h minutes before the 4:00 pm Fix. Notice
that this selection method does not require any estimation, so the returns I construct are from a strategy
that could be executed in real time. For the sake of comparison, I also construct returns from the same
strategy executed around all the intra-month Fixes.
I compute three performance measures to assess the attractiveness of the strategies to market participants:
(i) the average return, (ii) the Sharpe Ratio and (iii) the Maximum Drawdown. The Sharpe Ratio is
p
1
calculated as SR = p252
(ET [Ri ] 1) / Vt [R], where Ri is the (gross) return on day i. ET [.] and VT [R]
are sample the mean and variance from the T returns computed over the span of the data. Because returns
p
are generated at the daily frequency, I include the 1/ 252 scale factor to “annualize” the ratio (using
the convention that a year equals 252 trading days). Sharpe Ratios are widely used by financial market
participants to judge the attractiveness of trading strategies. The Maximum Drawdown statistic is another
widely-used measure. It is computed as the maximum percentage drop (i.e. from peak to trough) in the
cumulated return from following the trading strategy over the span of data. As such, it provides a measure
of downside risk.
Table 11 reports the performance measures for the trading strategies across all the currency pairs. The
returns from strategies executed at the end of each month are reported in Panel I, those from strategies
executed on intra-month days are shown in Panel II. Columns (i) - (iii) in Panel I show that average returns
are generally positive for the end-of-the-month strategies. For some currency pairs, the returns are above ten
percent (on an annualized basis). Average returns are also generally positive from the intra-month strategies
(see Panel II), but they are good deal smaller. The di↵erence between the end-of-month and intra-month
strategies carries over to the Sharpe Ratios. All the ratios from the intra-month strategies are below 2.6,
and most are below 2.0. Many of the Sharpe ratios from the end-of-month strategies are far higher, with a
few ranging above 5.0. By this metric, the intra-month strategies look much more attractive than the intramonth strategies. They also appear more attractive in terms of the Drawdown statistics. The Drawdowns in
the end-of-month strategies are generally one or two percent, whereas those from the intra-month strategies
range from two to almost 18 percent.
The results in Table 11 do not support the presence of a strong economic incentive to exploit rate
reversions around intra-month Fixes. Yes, the trading strategies for some currency pairs produce sizable
average returns (see, e.g. CAD/USD and NZD/GBP), but they are also very risky because the post-Fix
rate changes often di↵er from their forecast direction. Consequently, there does not appear to be a strong
incentive for market participants to enter into trades at the Fix in a manner that would further ameliorate
the temporal dependency between pre- and post-Fix rate changes observed in the intra-month data.
In contrast, there may be a stronger economic incentive to exploit the rate revisions around end-ofmonth Fixes. Panel I shows that strategies exploiting these rate reversions in many currency pairs produce
significantly higher average returns and Sharpe ratios and smaller Drawdown statistics. Trading around
the end-of-month Fixes appears to be more attractive than trading around the intra-month Fixes, but is it
attractive enough to produce an economic incentive to trade?
The answer to this question largely depends on the size of the trading costs. Table 11 reports performance
measures based on returns that use mid-point rates (i.e. the average of the bid and o↵er rates). As such,
41
42
4.113
5.151
4.153
15.149
7.755
8.120
-1.763
5.394
10.430
2.079
6.635
11.277
5.002
9.011
2.595
5.276
2.516
CHF
JPY
NOK
NZD
SEK
AUD
CAD
CHF
GBP
JPY
NZD
AUD
CAD
DKK
NOK
SEK
SGD
EUR
EUR
EUR
EUR
EUR
GBP
GBP
GBP
EUR
GBP
GBP
USD
USD
USD
USD
USD
USD
B:
C:
D:
14.382
11.987
3.603
6.245
-2.097
2.596
6.656
5.673
5.363
10.719
2.953
11.502
4.302
6.164
7.449
19.610
2.585
1.458
2.763
0.812
0.077
5
(ii)
10.443
10.907
1.680
10.719
3.667
0.719
3.133
1.402
1.637
8.761
0.613
10.890
3.698
3.001
4.806
6.963
4.502
1.040
4.582
0.233
-3.635
1
(iii)
4.133
1.700
3.484
0.753
1.231
2.177
2.506
-0.494
2.355
3.589
0.668
1.735
3.271
1.797
2.027
5.230
2.560
2.339
1.031
-0.066
-0.325
15
(iv)
5.086
4.334
1.854
2.012
-0.476
2.658
1.952
1.787
2.275
3.934
0.880
3.037
4.413
2.155
4.399
6.450
0.806
0.642
1.253
0.431
0.042
5
(v)
1
(vi)
3.904
4.108
1.018
5.169
0.993
0.841
1.448
0.530
0.927
3.237
0.256
4.476
5.158
1.250
2.738
2.741
1.687
0.488
2.197
0.151
-1.745
Sharpe Ratio
1.234
1.451
0.883
1.630
1.496
0.609
1.306
2.166
0.561
0.602
2.065
1.427
0.552
1.320
0.569
0.737
0.772
1.638
1.694
1.245
2.168
15
(vii)
0.923
0.977
0.893
1.008
2.892
0.304
0.982
1.427
0.732
0.439
1.788
1.291
0.363
0.930
0.343
0.529
1.006
1.694
1.077
1.356
1.475
5
(viii)
0.920
0.801
0.663
0.311
1.214
0.427
0.737
1.300
0.596
0.456
1.506
0.865
0.200
0.741
0.477
0.802
0.513
1.632
0.675
0.679
2.380
1
(ix)
Max Drawdown
0.200
5.006
0.760
3.851
3.567
1.546
1.847
3.701
2.218
1.751
-0.075
4.778
1.087
-1.161
3.735
3.973
2.774
-1.732
0.854
0.481
-0.655
15
(i)
0.535
5.349
0.665
3.119
2.207
1.914
2.472
4.333
3.075
2.282
1.061
6.356
1.725
-0.053
3.370
4.682
2.761
-1.013
1.594
0.504
0.192
5
(ii)
1.006
3.845
0.736
2.964
0.278
2.205
2.041
3.420
2.347
1.674
0.974
5.862
1.606
-0.875
2.664
4.782
0.128
-1.686
1.273
0.992
0.254
1
(iii)
Average Return
0.091
2.275
0.344
1.215
1.148
1.427
0.735
1.542
1.152
1.191
-0.017
1.794
0.799
-0.469
2.088
1.605
1.547
-0.854
0.401
0.280
-0.313
15
(iv)
0.240
2.635
0.328
1.081
0.744
2.015
1.083
1.975
1.759
1.676
0.468
2.533
1.423
-0.014
1.959
2.019
1.698
-0.554
0.800
0.336
0.120
5
(v)
1
(vi)
0.465
2.032
0.373
1.124
0.113
2.386
0.953
1.686
1.429
1.312
0.475
2.389
1.452
-0.397
1.657
2.173
0.091
-0.980
0.719
0.667
0.165
Sharpe Ratio
II: Intra-month
12.621
3.992
4.303
2.748
3.868
1.155
4.178
3.722
4.991
2.046
16.879
2.653
2.696
17.847
1.479
2.919
3.114
17.677
5.724
9.225
13.563
15
(vii)
11.112
2.515
5.188
5.158
4.621
1.263
2.432
1.945
2.228
1.361
6.396
2.796
2.359
6.502
2.303
3.151
1.688
13.217
3.993
5.489
6.492
5
(viii)
7.103
3.832
6.624
3.826
7.964
0.730
3.035
2.468
2.404
1.634
3.492
2.436
2.259
10.494
2.111
3.322
2.513
16.370
3.890
4.955
6.165
1
(ix)
Max Drawdown
Notes: Columns (i) - (iii) report the average return (in annual percent) from a trading strategy of holding a long (short) position for horizon h = {1, 5, 15} minutes following the Fix if
the Fix is below (above) the price level h minutes earlier. Columns (iv) - (vi) report the associated Sharpe ratios (annualized), while columns (vii) - (ix) show the maximum drawdown
in percent from following the strategy on every end-of-month trading day (Panel I) and every intra-month trading day (Panel II).
4.937
2.921
-0.167
-0.866
EUR USD
USD CHF
USD JPY
GBP USD
15
(i)
A:
Horizon
Average Return
I: End-of-month
Table 11: Trading Around the Fix
they do not include the trading costs of entering a position at the Fix and exiting some minutes later. In
reality, spreads collapse to almost zero in the 60-second window around 4:00 pm used in computing the Fix,
so the Fix benchmark is a good approximation to the transaction price that traders would actually face
when initiating a position at 4:00 pm. Thereafter spreads return to their normal level for the 20-30 minutes
until daily trading activity declines. This pattern suggests that the typical rate facing a trader closing out
a position from one to fifteen minutes after the Fix would be equal to the mid-point rate ± one half the
normal spread between the o↵er and bid rates.
Table 12: Trading Around the Fix with Transaction Costs
Average Return
Horizon
Sharpe Ratio
Drawdown
Spread
15
(i)
5
(ii)
1
(iii)
15
(iv)
5
(v)
1
(vi)
15
(vii)
5
(viii)
1
(ix)
(Basis Points)
(x)
A:
EUR USD
USD CHF
USD JPY
GBP USD
2.807
-1.358
-3.699
-3.650
-0.673
-1.515
-2.720
-2.682
-1.090
0.303
-3.272
-6.395
1.335
-0.458
-1.694
-1.415
-0.279
-0.669
-1.399
-0.977
-0.489
0.155
-2.000
-3.079
1.998
1.935
1.799
3.007
1.931
1.273
1.835
1.858
1.852
0.871
1.486
3.321
1.708
3.477
2.771
2.285
B:
EUR
EUR
EUR
EUR
EUR
CHF
JPY
NOK
NZD
SEK
1.402
1.959
-1.360
6.247
3.351
1.636
2.972
2.029
10.585
-1.818
1.077
-0.191
-0.706
-2.062
0.097
1.121
0.693
-0.650
2.157
1.115
1.684
1.047
1.205
3.467
-0.540
1.511
-0.067
-0.393
-0.792
0.049
0.637
1.615
0.828
0.951
0.856
0.464
1.207
0.482
0.618
1.461
0.353
1.039
0.577
1.711
0.764
2.160
2.622
4.449
7.018
3.584
C:
GBP
GBP
GBP
EUR
GBP
GBP
AUD
CAD
CHF
GBP
JPY
NZD
2.244
-7.687
0.242
6.221
-3.089
-5.669
0.781
-0.252
0.267
6.510
-2.216
-0.600
-2.746
-4.362
-3.353
4.552
-4.556
-1.420
0.701
-2.212
0.117
2.142
-0.953
-1.442
0.244
-0.063
0.125
2.391
-0.631
-0.138
-1.244
-1.595
-1.867
1.685
-1.795
-0.565
2.074
3.109
1.745
1.184
3.385
3.480
1.016
1.862
1.122
0.905
2.694
2.281
1.665
1.728
2.011
0.810
3.278
1.854
4.773
4.841
4.152
3.208
4.090
9.738
D:
USD
USD
USD
USD
USD
USD
AUD
CAD
DKK
NOK
SEK
SGD
7.097
0.576
7.416
-3.413
0.122
-1.898
10.202
7.679
2.007
0.236
-7.250
-1.891
6.263
6.550
0.085
4.709
-1.487
-3.490
2.614
0.209
2.869
-0.950
0.049
-1.626
3.619
2.780
1.037
0.091
-1.699
-1.920
2.345
2.470
0.059
2.276
-0.377
-4.040
1.828
2.339
0.987
2.435
1.969
1.092
1.472
1.089
1.116
1.631
3.673
0.813
1.521
0.913
0.810
0.429
1.833
0.991
3.171
3.576
1.244
4.738
4.048
3.671
Notes: Columns (i) - (iii) report the average return (in annual percent) from a trading strategy of holding a long (short)
position for horizon h = {1, 5, 15} minutes following the end-of-month Fix if the Fix is below (above) the price level h minutes
earlier. Columns (iv) - (vi) report the associated Sharpe ratios (annualized), while columns (vii) - (ix) show the maximum
drawdown in percent from following the strategy on every end-of-month trading day. Returns are inclusive of trading costs,
computed to be zero at the Fix and one half the average bid-ask spread (shown in column x) when the position is closed.
Table 12 reports the performance measures for the end-of-month trading strategy that include a trading
cost of half the average spread estimated between 7:00 am and 6:00 pm GMT on every day in the data span.
As the table clearly shows, the inclusion of this trading cost has a significant impact on the performance
measures. Average returns are considerably lower; indeed, for many currency pairs they are now below zero.
There are, however, a number of cases where average returns remain large a positive. For example, returns
for the JPY/EUR, NZD/EUR, EUR/GBP, AUD/USD, CAD/USD, NOK/USD and DKK/USD at one or
more horizons are sizable. The Sharpe Ratios and Drawdown statistics also appear quite attractive in many
of these currencies.
43
The di↵erence between the performance measures for the end-of-month strategies in Tables 11 and 12
show that the strength of the economic incentive to exploit rate revisions around end-of-month Fixes depends
critically on trading costs. These costs di↵er from one market participant to another according to the trading
venues they use, so it is impossible to compute a single performance measure (inclusive of trading costs) that
is relevant to every market participant. Undoubtedly, some participants have access to trading platforms
where spreads are much smaller than the average spreads reported in the Table 12. These participants
face stronger economic incentives to exploit the rate revisions around the end-of-month Fixes than the
performance measures in Table 12 suggest. For others, facing larger costs, the incentives are far weaker.
Indeed, the performance measures in Table 12 indicate that they are absent for many of the currency pairs.
In summary, the performance metrics in Tables 11 and 12 suggest that for some currency pairs, most
notably the NZD/EUR, EUR/GBP, AUD/USD and CAD/USD, market participants face strong economic
incentives to adopt trading strategies exploiting rate revisions around end-of-month Fixes. For other currency
pairs (including the four majors), the economic incentives are less clear cut because the metrics are far more
sensitive to trading costs.
8
Conclusion
This paper has documented the atypical behavior of forex spot rates around the 4:00 pm Fix, particularly
at the end of each month. The results show that across all time periods and currency pairs changes in
rates before and after the Fix are regularly of a size rarely seen in normal trading activity. The pre- and
post-Fix rate changes also display a strong degree of negative autocorrelation that is not found elsewhere
during normal forex trading. As a consequence, there appears to be a strong economic incentive for market
participant to adopt trading strategies that exploit the implied reversion in the rates (for some currency
pairs) around the Fix.
These findings represent a challenge to standard forex trading models. Because the Fix is used in the
real-time valuation of financial benchmarks and contracts, there is clear hedging motive to execute forex
transaction at the Fix. Consequently, it is not a surprise that forex rates are unusually volatile in the 60second Fix window around 4:00 pm. According to standard trading models (like the PS model discussed in
Section 1), this is the period where rates should adjust to (unanticipated) aggregate market-wide order flow
generated by hedging forex trades. What is surprising is the scale and timing. Volatility is so much higher
than observed at other times, and rates start jumping around well before the Fix window. Standard trading
models can only account for this level of volatility in the presence of very large (unanticipated) order flows,
and cannot predict the anticipatory movements in the rates before the Fix. Also, the models cannot account
for the strong negative correlation in rate changes around the Fix that appear to present attractive trading
opportunities.
How, then, should we interpret these findings, particularly the autocorrelation in spot rate changes
around the Fix? One possibility is simply that market participants were unaware of the trading opportunity
it represented, but this not a compelling explanation. A disproportionately large amount of daily trading
volume takes place during the minute or so around the Fix (approximately one percent of daily volume),
so one would expect that many market participants focus on the behavior of spot rates during this period.
44
Alternatively, participants could have been aware of the trading opportunity, and (some) were exploiting
it, but the e↵ect of their trades on rates was o↵set be another countervailing factor. This seems a more
plausible explanation, but it is impossible to investigate it further without detailed data on trading activity
around the Fix.
References
Bowman, Adrian W and Adelchi Azzalini. 1997. Applied Smoothing Techniques for Data Analysis: The Kernel Approach with S-Plus Illustrations: The Kernel Approach with S-Plus Illustrations. Oxford University
Press. 6
Evans, Martin D. D. 2011. Exchange-Rate Dynamics. Princeton Series in International Finance. Princeton
University Press. 1.2
Evans, Martin D.D. and Richard K. Lyons. 2002. “Order flow and exchange rate dynamics.” Journal of
political economy 110 (1):170–180. 1.2
Lyons, Richard K. 1997. “A Simultaneous Trade Model of the Foreign Exchange Hot Potato.” Journal of
International Economics 42 (3-4):275–298. 1.2
Melvin, Michael and John Prins. 2011. “The Equity Hedging Channel of Exchange Rate Adjustment.” Tech.
rep., Blackrock. (document), 1.1
45
Appendix to Forex Trading and the WMR Fix
Martin D. D. Evans
27th August 2014
1
CHF/EUR
DKK/EUR
JPY/EUR
NOK/EUR
NZD/EUR
SEK/EUR
Average1
AUS/GBP
CAD/GBP
CHF/GBP
EUR/GBP
JPY/GBP
NZD/GBP
Average
AUS/USD
CAD/USD
DKK/USD
HKD/USD
NOK/USD
SEK/USD
SGD/USD
Average2
B: EUR
C: GBP
D: USD
85.759
85.643
85.361
3.223
115.907
112.579
37.107
87.059
91.081
96.104
75.872
63.237
95.379
92.833
85.751
35.235
2.351
82.624
69.163
87.416
61.625
67.213
76.794
94.853
73.341
81.742
81.683
175.769
153.332
129.650
10.648
184.613
192.450
63.284
149.850
167.789
167.363
143.806
119.544
160.330
155.445
152.380
90.774
4.835
154.654
110.285
150.125
127.977
126.763
119.249
154.358
119.288
135.705
132.150
(ii)
0.246
0.225
0.224
0.330
0.232
0.275
0.273
0.246
0.268
0.361
0.226
0.181
0.339
0.286
0.277
0.272
0.270
0.243
0.258
0.321
0.171
0.253
0.268
0.319
0.312
0.286
0.296
(iii)
0.150
0.141
0.132
0.248
0.154
0.165
0.191
0.155
0.154
0.192
0.165
0.116
0.200
0.174
0.167
0.149
0.166
0.154
0.176
0.216
0.065
0.152
0.167
0.183
0.142
0.160
0.163
(iv)
46.425
47.143
39.292
2.314
58.704
63.217
18.979
45.627
47.206
50.003
34.745
29.016
47.713
52.234
43.486
19.247
1.409
43.303
33.250
49.377
37.374
36.510
38.834
47.528
37.440
41.052
41.213
(i)
91.047
76.322
73.290
7.958
92.952
116.613
32.221
80.407
86.571
99.829
79.824
66.500
101.746
92.520
87.832
45.610
3.087
91.082
61.362
94.327
75.349
73.546
75.895
81.867
76.826
78.282
78.217
(ii)
0.280
0.236
0.289
0.333
0.309
0.360
0.340
0.302
0.312
0.282
0.329
0.266
0.322
0.337
0.308
0.267
0.218
0.300
0.347
0.323
0.192
0.286
0.351
0.367
0.341
0.405
0.366
(iii)
0.153
0.164
0.174
0.211
0.187
0.208
0.231
0.186
0.174
0.187
0.253
0.152
0.220
0.206
0.199
0.189
0.062
0.158
0.231
0.259
0.125
0.192
0.177
0.163
0.203
0.251
0.199
(iv)
II: 3:00-5:00 GMT
Range Distribution Tail Probabilities
50%
90%
20%
10%
33.151
32.877
29.290
2.058
42.882
43.960
14.129
32.715
39.020
38.543
27.241
23.569
34.551
41.126
34.008
14.603
1.072
32.059
24.722
36.202
30.441
27.605
28.676
31.707
27.451
30.964
29.700
(i)
65.346
61.621
52.409
6.274
82.140
90.279
23.480
62.546
71.700
80.019
66.374
46.696
85.630
75.692
71.019
35.722
2.422
76.211
48.031
70.396
56.439
57.360
52.211
64.957
55.131
68.833
60.283
(ii)
0.244
0.228
0.273
0.270
0.306
0.287
0.386
0.287
0.289
0.268
0.297
0.219
0.309
0.376
0.293
0.265
0.226
0.288
0.318
0.315
0.213
0.280
0.313
0.361
0.308
0.309
0.323
(iii)
0.135
0.174
0.183
0.185
0.170
0.217
0.280
0.193
0.173
0.202
0.176
0.099
0.181
0.268
0.183
0.170
0.081
0.174
0.192
0.194
0.143
0.175
0.164
0.204
0.214
0.214
0.199
(iv)
III: 3:30-4:30 GMT
Range Distribution
Tail Probabilities
50%
90%
20%
10%
points; i.e., (ln(P h ) ln(P l ))10000 where P h and P l are the highest and lowest quotes (midpoint of bid and ask) within the range. Column (iii) report the fraction of days in the sample that the
P l )/(P h P l ) is either below 0.1 or above 0.9. Column (iv) reports the fraction of the days when the ratio is either below 0.05 or above 0.95. Averages for the currencies in each
ratio (P f
block are reported in the last row (1: excludes DKK/EUR, 2: excludes HKD/USD).
Notes: Columns (i) and (ii) report the 50th. and 90th. percentiles from the empirical distribution of the end-of-month trading range (identified in the header of each panel) expressed in basis
EUR/USD
CHF/USD
JPY/USD
USD/GBP
Average
A: Majors
(i)
I: 7:00-6:00 GMT
Range Distribution Tail Probabilities
50%
90%
20%
10%
Table A.1: End-Of-Month Trading Ranges and the Fix
2
CHF/EUR
DKK/EUR
JPY/EUR
NOK/EUR
NZD/EUR
SEK/EUR
Average1
AUS/GBP
CAD/GBP
CHF/GBP
EUR/GBP
JPY/GBP
NZD/GBP
Average
AUS/USD
CAD/USD
DKK/USD
HKD/USD
NOK/USD
SEK/USD
SGD/USD
Average2
B: EUR
C: GBP
D: USD
78.026
73.848
80.100
2.448
105.399
110.103
36.732
80.701
79.466
81.295
65.698
57.021
80.681
86.155
75.053
32.896
1.884
79.043
61.213
81.948
65.548
64.130
72.664
78.339
66.044
68.381
71.357
160.820
137.284
148.223
5.911
197.682
209.408
68.278
153.616
155.353
152.244
133.619
111.370
165.320
162.276
146.697
91.019
3.871
163.981
122.101
151.939
129.019
131.612
133.544
141.763
121.030
128.790
131.282
(ii)
0.335
0.288
0.306
0.264
0.314
0.300
0.315
0.310
0.295
0.284
0.290
0.249
0.292
0.296
0.284
0.346
0.358
0.301
0.272
0.304
0.265
0.297
0.305
0.321
0.304
0.280
0.302
(iii)
0.225
0.183
0.219
0.146
0.200
0.192
0.188
0.201
0.205
0.176
0.191
0.157
0.177
0.189
0.182
0.225
0.226
0.196
0.163
0.206
0.157
0.189
0.212
0.219
0.200
0.177
0.202
(iv)
37.366
34.708
36.904
1.290
49.816
51.721
16.816
37.889
36.084
38.477
28.449
23.384
34.177
41.277
33.641
15.084
0.938
34.769
28.638
38.488
29.633
29.322
32.311
35.631
29.283
29.223
31.612
(i)
80.620
69.904
70.222
3.337
95.061
97.139
31.438
74.064
73.708
75.942
57.954
46.463
75.072
80.718
68.310
41.673
2.012
73.595
54.529
76.247
56.571
60.523
64.024
68.119
59.179
57.764
62.272
(ii)
0.376
0.332
0.415
0.247
0.349
0.354
0.344
0.362
0.365
0.317
0.359
0.334
0.349
0.336
0.343
0.335
0.297
0.369
0.275
0.345
0.287
0.322
0.411
0.396
0.376
0.357
0.385
(iii)
0.235
0.205
0.279
0.140
0.221
0.213
0.222
0.229
0.233
0.204
0.218
0.195
0.233
0.204
0.215
0.209
0.117
0.243
0.167
0.205
0.177
0.200
0.276
0.256
0.244
0.230
0.251
(iv)
II: 3:00-5:00 GMT
Range Distribution Tail Probabilities
50%
90%
20%
10%
26.729
24.513
25.051
1.029
35.412
36.153
11.499
26.560
26.107
27.691
20.563
16.814
24.465
30.054
24.282
10.994
0.671
23.843
20.576
28.206
21.945
21.113
22.000
24.300
20.271
20.340
21.728
(i)
55.073
48.278
49.690
2.570
66.993
69.350
23.373
52.126
56.145
55.438
42.024
33.479
52.827
61.080
50.166
30.413
1.612
51.180
41.120
56.326
41.317
44.071
44.306
47.568
39.128
41.082
43.021
(ii)
0.340
0.305
0.403
0.225
0.333
0.332
0.301
0.336
0.339
0.307
0.333
0.305
0.325
0.293
0.317
0.317
0.182
0.367
0.244
0.313
0.254
0.299
0.395
0.360
0.348
0.339
0.360
(iii)
0.204
0.188
0.268
0.094
0.203
0.205
0.189
0.209
0.205
0.210
0.209
0.177
0.213
0.197
0.202
0.186
0.111
0.224
0.154
0.197
0.168
0.186
0.253
0.234
0.235
0.213
0.234
(iv)
III: 3:30-4:30 GMT
Range Distribution
Tail Probabilities
50%
90%
20%
10%
block are reported in the last row (1: excludes DKK/EUR, 2: excludes HKD/USD).
Notes: Columns (i) and (ii) report the 50th. and 90th. percentiles from the empirical distribution of the intra-month trading range (identified in the header of each panel) expressed in basis
points; i.e., (ln(P h ) ln(P l ))10000 where P h and P l are the highest and lowest quotes (midpoint of bid and ask) within the range. Column (iii) report the fraction of days in the sample that the
P l )/(P h P l ) is either below 0.1 or above 0.9. Column (iv) reports the fraction of the days when the ratio is either below 0.05 or above 0.95. Averages for the currencies in each
ratio (P f
EUR/USD
CHF/USD
JPY/USD
USD/GBP
Average
A: Majors
(i)
I: 7:00-6:00 GMT
Range Distribution Tail Probabilities
50%
90%
20%
10%
Table A.2: Intra-Month Trading Ranges and the Fix
3
09
05
06
10
07
08
11
NOK/EUR
09
CHF/EUR
12
10
11
13
12
13
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
90
100
110
120
130
140
150
160
170
Notes: Time series for the fix at the end of each month with upper and lower limits of daily trading range.
7
7.5
8
8.5
9
9.5
10
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
05
09
06
Figure A.1: Fixes with Daily Trading Range
10
07
08
11
NZD/EUR
09
JPY/EUR
10
12
11
12
13
13
4
09
10
11
12
13
1.4
1.6
1.8
2
2.2
2.4
2.6
Notes: Time series for the fix at the end of each month with upper and lower limits of daily trading range.
1.5
1.6
1.7
1.8
1.9
2
2.1
2.8
1.4
8
CAD/GBP
1.6
8.5
2.2
1.8
9
13
2
9.5
12
2.2
10
11
2.4
10.5
10
2.6
11
09
2.8
11.5
SEK/EUR
05
09
06
Figure A.2: Fixes with Daily Trading Range
07
10
08
09
CHF/GBP
11
AUD/GBP
10
12
11
12
13
13
5
05
09
06
10
07
08
11
NZD/GBP
09
GBP/EUR
10
12
11
12
13
13
0.7
0.8
0.9
1
1.1
1.2
1.3
100
150
200
250
Notes: Time series for the fix at the end of each month with upper and lower limits of daily trading range.
1.8
2
2.2
2.4
2.6
2.8
3
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
05
05
06
06
Figure A.3: Fixes with Daily Trading Range
07
07
08
08
09
AUD/USD
09
JPY/GBP
10
10
11
11
12
12
13
13
6
09
05
06
10
07
08
11
NOK/USD
09
CAD/USD
12
10
11
13
12
13
5.5
6
6.5
7
7.5
8
8.5
9
9.5
5
5.2
5.4
5.6
5.8
6
6.2
6.4
09
09
Notes: Time series for the fix at the end of each month with upper and lower limits of daily trading range.
5.2
5.4
5.6
5.8
6
6.2
6.4
6.6
6.8
7
7.2
0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4
Figure A.4: Fixes with Daily Trading Range
10
10
11
11
SEK/USD
DKK/USD
12
12
13
13
7
09
10
11
SGD/USD
12
13
Notes: Time series for the fix at the end of each month with upper and lower limits of daily trading range.
1.25
1.3
1.35
1.4
1.45
1.5
1.55
1.6
Figure A.5: Fixes with Daily Trading Range
8
9
09
05
06
10
07
08
11
NOK/EUR
09
CHF/EUR
12
10
11
13
12
13
−150
−100
−50
0
50
100
150
200
250
300
−250
−200
−150
−100
−50
0
50
100
150
200
250
Notes: Time series for the fix at the end of each month with upper and lower limits of daily trading range.
−200
−150
−100
−50
0
50
100
150
200
−200
−150
−100
−50
0
50
100
150
200
05
09
06
Figure A.6: Daily Trading Range Around Fix
10
07
08
11
NZD/EUR
09
JPY/EUR
10
12
11
12
13
13
10
09
10
11
12
13
−200
−150
−100
−50
0
50
100
150
200
250
Notes: Time series for the fix at the end of each month with upper and lower limits of daily trading range.
−300
−200
−100
0
100
200
300
−300
−200
CAD/GBP
−200
−150
300
−100
−100
13
0
−50
12
100
0
11
200
50
10
300
100
09
400
150
SEK/EUR
05
09
06
Figure A.7: Daily Trading Range Around Fix
07
10
08
09
CHF/GBP
11
AUD/GBP
10
12
11
12
13
13
11
13
Notes: Time series for the fix at the end of each month with upper and lower limits of daily trading range.
−400
12
−300
11
−300
−200
10
−200
−100
−250
−100
13
0
12
0
11
100
10
100
NZD/GBP
09
200
08
200
09
07
300
06
−200
−150
−100
−50
0
50
100
150
200
250
300
05
GBP/EUR
400
−200
−150
−100
−50
0
50
100
150
200
05
05
06
06
Figure A.8: Daily Trading Range Around Fix
07
07
08
08
09
AUD/USD
09
JPY/GBP
10
10
11
11
12
12
13
13
12
09
05
06
10
07
08
11
NOK/USD
09
CAD/USD
12
10
11
13
12
13
−400
−300
−200
−100
0
100
200
300
−300
−250
−200
−150
−100
−50
0
50
100
150
200
09
09
Notes: Time series for the fix at the end of each month with upper and lower limits of daily trading range.
−300
−250
−200
−150
−100
−50
0
50
100
150
200
−200
−150
−100
−50
0
50
100
150
200
10
10
Figure A.9: Daily Trading Range Around Fix
11
11
SEK/USD
DKK/USD
12
12
13
13
13
09
10
11
SGD/USD
12
13
Notes: Time series for the fix at the end of each month with upper and lower limits of daily trading range.
−100
−50
0
50
100
150
Figure A.10: Daily Trading Range Around Fix
14
−2
−2
−2
0
iii: NOK/EUR
0
i: NOK/EUR
0
2
2
2
4
4
4
0
−4
0.5
1
1.5
0
−4
0.5
1
1.5
−2
−2
−2
0
iv: NOK/EUR
0
iv: CHF/EUR
0
ii: CHF/EUR
2
2
2
4
4
4
0
−4
0.5
1
1.5
2
0
−4
0.5
1
1.5
2
0
−4
0.5
1
1.5
2
0
−4
0.5
1
1.5
2
−2
−2
−2
−2
0
iii: NZD/EUR
0
i: NZD/EUR
0
iii: JPY/EUR
0
i: JPY/EUR
2
2
2
2
4
4
4
4
0
−4
0.2
0.4
0.6
0.8
1
0
−4
0.2
0.4
0.6
0.8
0
−4
0.5
1
1.5
2
2.5
−2
−2
−2
0
iv: NZD/EUR
0
iv: JPY/EUR
0
ii: JPY/EUR
2
2
2
4
4
4
Notes: Panel i plots the density functions for h st for h = {5, 15, 30} minutes in green, blue, and red, respectively. Panel ii plots the density functions h st from pre-2008 and
post 2009 data with solid and dotted lines, respectively. Panels iii and iv plod the conditional densities for f ( h st | h st h > + ) (solid) and f ( h st | h st h <  ) (dotted) for
{+ ,  }= {75%, 25%} (panel iii) and {97.5%, 2.5%} (panel iv).
0
−4
0.5
1
1.5
2
2.5
3
0
−4
0.5
1
1.5
2
2.5
0
−4
1
2
3
4
iii: CHF/EUR
0
−4
0
−4
5
1
1
4
2
2
2
3
3
0
4
4
−2
5
i: CHF/EUR
5
Figure A.11: Rate Change Densities
15
−2
−2
0
iii: CAD/GBP
0
i: CAD/GBP
2
2
4
4
0
−4
0.2
0.4
4
0
−4
0.5
1
1.5
0.6
0
−4
0.5
1
4
−2
0
iii: CHF/GBP
2
4
0
−4
0.2
0.4
0.6
0.8
0
−4
0.5
1
1.5
0
−4
0.2
0.4
0.6
0.8
1.5
2
4
4
2
0
2
2
2
i: CHF/GBP
0
iii: AUD/GBP
0
2.5
−2
−2
−2
i: AUD/GBP
2.5
0
−4
2
2
4
0.8
0
2
2.5
iv: CAD/GBP
0
0.5
1
1.5
2
1
−2
−2
iv: SEK/EUR
0
−4
0.5
1
1.5
2
−2
−2
−2
0
iv: CHF/GBP
0
ii: CHF/GBP
0
iv: AUD/GBP
2
2
2
4
4
4
Notes: Panel i plots the density functions for h st for h = {5, 15, 30} minutes in green, blue, and red, respectively. Panel ii plots the density functions h st from pre-2008 and
post 2009 data with solid and dotted lines, respectively. Panels iii and iv plod the conditional densities for f ( h st | h st h > + ) (solid) and f ( h st | h st h <  ) (dotted) for
{+ ,  }= {75%, 25%} (panel iii) and {97.5%, 2.5%} (panel iv).
0
−4
0.5
1
1.5
2
0
−4
0.5
1
1.5
2
0
−4
0.2
0.5
0
−4
0.4
1
1
4
4
0.6
2
2
0.8
0
iii: SEK/EUR
0
2
−2
−2
i: SEK/EUR
1.5
2.5
0
−4
0.5
1
1.5
2
2.5
Figure A.12: Rate Change Densities
16
2
4
−2
−2
−2
0
iv: NZD/GBP
0
iv: GBP/EUR
0
ii: GBP/EUR
2
2
2
4
4
4
2
4
4
4
0
−4
0.2
0.4
0.6
0.8
0
−4
0
−4
0.5
1
1.5
2
0
−4
0.5
1
−2
0
iii: AUD/USD
2
4
0
−4
0.2
0.4
0.6
0.8
0
−4
0.5
1
1.5
0
2
2
1.5
i: AUD/USD
0
iii: JPY/GBP
0
0.5
1
1.5
2
2
−2
−2
−2
i: JPY/GBP
2
0
−4
0.5
1
1.5
2
0
−4
0.5
1
1.5
2
−2
−2
−2
−2
0
iv: AUD/USD
0
ii: AUD/USD
0
iv: JPY/GBP
0
ii: JPY/GBP
2
2
2
2
4
4
4
4
Notes: Panel i plots the density functions for h st for h = {5, 15, 30} minutes in green, blue, and red, respectively. Panel ii plots the density functions h st from pre-2008 and
post 2009 data with solid and dotted lines, respectively. Panels iii and iv plod the conditional densities for f ( h st | h st h > + ) (solid) and f ( h st | h st h <  ) (dotted) for
{+ ,  }= {75%, 25%} (panel iii) and {97.5%, 2.5%} (panel iv).
0
−4
0.2
0.5
0
−4
0.4
0
−4
0.5
1
1.5
0
−4
1
0
4
4
4
0.6
−2
2
2
2
1.5
iii: NZD/GBP
0
i: NZD/GBP
0
iii: GBP/EUR
0
0.5
1
1.5
2
2.5
3
0.8
−2
−2
−2
i: GBP/EUR
2
0
−4
0.5
1
1.5
0
−4
0.5
1
1.5
2
2.5
3
0
−4
0.5
1
1.5
2
2.5
Figure A.13: Rate Change Densities
17
4
4
−2
−2
0
iii: NOK/USD
0
i: NOK/USD
2
2
4
4
4
0
−4
0.5
1
1.5
2
0
−4
0.5
1
1.5
2
0
−4
0
−4
0.2
0.4
0
−4
0.5
1
1.5
2
4
4
0.6
0
2
2
2
iv: NOK/USD
0
iv: CAD/USD
0
0.5
1
1.5
2
0.8
−2
−2
−2
ii: CAD/USD
−2
−2
−2
−2
0
iii: CHF/USD
0
i: CHF/USD
0
iii: DKK/USD
0
i: DKK/USD
2
2
2
2
4
4
4
4
0
−4
0.2
0.4
0.6
0.8
1
0
−4
0.5
1
1.5
2
2.5
0
−4
0.2
0.4
0.6
0.8
1
−2
−2
−2
0
iv: CHF/USD
0
ii: CHF/USD
0
iv: DKK/USD
2
2
2
4
4
4
Notes: Panel i plots the density functions for h st for h = {5, 15, 30} minutes in green, blue, and red, respectively. Panel ii plots the density functions h st from pre-2008 and
post 2009 data with solid and dotted lines, respectively. Panels iii and iv plod the conditional densities for f ( h st | h st h > + ) (solid) and f ( h st | h st h <  ) (dotted) for
{+ ,  }= {75%, 25%} (panel iii) and {97.5%, 2.5%} (panel iv).
0
−4
0.5
1
1.5
0
−4
0.5
1
1.5
0
−4
0.2
0.5
2
0.4
1
0
0.6
0
−4
0.8
1
2
−2
iii: CAD/USD
1.5
2.5
0
−4
0.5
0.5
2
1
1
0
1.5
1.5
−2
2
2
0
−4
2.5
i: CAD/USD
2.5
Figure A.14: Rate Change Densities
18
4
−2
0
iv: SGD/USD
2
4
Notes: Panel i plots the density functions for h st for h = {5, 15, 30} minutes in green, blue, and red, respectively. Panel ii plots the density functions h st from pre-2008 and
post 2009 data with solid and dotted lines, respectively. Panels iii and iv plod the conditional densities for f ( h st | h st h > + ) (solid) and f ( h st | h st h <  ) (dotted) for
{+ ,  }= {75%, 25%} (panel iii) and {97.5%, 2.5%} (panel iv).
0
−4
0.5
1
0
−4
1
2
2
4
1.5
0
2
3
iii: SGD/USD
0
2
−2
−2
i: SGD/USD
4
0
−4
1
2
3
4
Figure A.15: Rate Change Densities
19
0
CHF/EUR 5 mins
0
10
50
10
20
0
10
−10
0
NOK/EUR 1 min
0
10
NOK/EUR 15 mins
−10
CHF/EUR 1 min
0
CHF/EUR 15 mins
20
50
20
50
0
−20
0.02
0.04
20
0
−20
0.05
0.1
0.2
0
−50
0.02
0.04
0.06
0.08
0
−20
0.05
0.1
0.15
0.2
0.25
0
−50
0.15
10
100
20
100
0.06
0
50
10
50
0.08
NZD/EUR 5 mins
0
NZD/EUR 60 mins
0
JPY/EUR 5 mins
0
0.02
0.04
0.06
0.08
0.25
−10
−50
−10
−50
JPY/EUR 60 mins
0.1
0
−100
0.01
0.02
0.03
0.04
0
−20
0.02
0.04
0.06
0.08
0.1
0.12
0
−100
0.01
0.02
0.03
0.04
Notes: Densities of price changes (in basis points) away from Fix (black) intra-month pre-Fix (blue) and end-of-month pre-Fix (red).
0
−20
0.2
0.05
0
−20
0.4
0
−50
0.1
0
100
0.6
−10
50
0.02
0.04
0.06
0.08
0.1
0
−20
0.1
0.2
0.3
0.4
0.5
0
−50
0.15
NOK/EUR 5 mins
0
20
100
0.05
0.1
0.15
0.2
0.8
−50
NOK/EUR 60 mins
−10
−50
CHF/EUR 60 mins
0.2
0
−100
0.01
0.02
0.03
0.04
0
−20
0.1
0.2
0.3
0.4
0
−100
0.02
0.04
0.06
0.08
0.1
Figure A.16: Pre-Fix Rate Change Densities
−10
−10
0
NZD/EUR 1 min
0
NZD/EUR 15 mins
0
JPY/EUR 1 min
0
JPY/EUR 15 mins
10
10
20
50
20
50
20
0
SEK/EUR 5 mins
0
10
50
−10
−50
0
CAD/GBP 5 mins
0
10
50
CAD/GBP 60 mins
−10
−50
SEK/EUR 60 mins
20
100
20
100
0
−20
0.05
0.1
0.15
0.2
0.25
0
−50
0.02
0.04
0.06
0.08
0
−20
0.1
10
−10
0
CAD/GBP 1 min
0
10
CAD/GBP 15 mins
0
20
20
50
0
−20
0.05
0.1
0.15
0.2
0
−100
0.01
0.02
0.03
0.04
0
−20
0.02
0.04
0.06
0.3
0.2
0.08
0
−100
0.4
−10
50
0.01
0.02
0.03
0.04
0.1
SEK/EUR 1 min
0
SEK/EUR 15 mins
0.5
0
−50
0.02
0.04
0.06
0.08
0.1
−10
−50
−10
−50
0
CHF/GBP 5 mins
0
CHF/GBP 60 mins
0
AUD/GBP 5 mins
0
10
50
10
50
AUD/GBP 60 mins
20
100
20
100
Notes: Densities of price changes (in basis points) away from Fix (black) intra-month pre-Fix (blue) and end-of-month pre-Fix (red).
0
−20
0.02
0.04
0.06
0.08
0.1
0.12
0
−100
0.005
0.01
0.015
0.02
0.025
0.03
0
−20
0.05
0.1
0.15
0.2
0
−100
0.01
0.02
0.03
0.04
Figure A.17: Pre-Fix Rate Change Densities
0
−20
0.1
0.2
0.3
0.4
0
−50
0.02
0.04
0.06
0.08
0.1
0
−20
0.05
0.1
0.15
0.2
0.25
0
−50
0.02
0.04
0.06
0.08
−10
−10
0
CHF/GBP 1 min
0
CHF/GBP 15 mins
0
AUD/GBP 1 min
0
10
10
AUD/GBP 15 mins
20
50
20
50
21
100
10
20
−10
0
NZD/GBP 5 mins
10
20
0
−20
0.05
0.1
0.15
0.2
−10
−10
0
NZD/GBP 1 min
0
NZD/GBP 15 mins
0
GBP/EUR 1 min
0
10
10
GBP/EUR 15 mins
20
50
20
50
0
−20
0.02
0.04
0.06
0.08
0.1
0.12
0
−100
0.01
0.02
0.03
0.04
0
−20
0.02
0.04
0.06
0.08
0.1
0.12
0
−100
0.01
0.02
0.03
0.04
−10
−50
−10
−50
10
50
0
AUD/USD 5 mins
0
10
50
AUD/USD 60 mins
0
JPY/GBP 5 mins
0
JPY/GBP 60 mins
20
100
20
100
Notes: Densities of price changes (in basis points) away from Fix (black) intra-month pre-Fix (blue) and end-of-month pre-Fix (red).
0
−20
0.02
0.04
0.06
0.08
0.1
0
−50
0.01
0.005
100
0.02
0.01
50
0.03
0.015
0
0.04
0.02
−50
0.05
0.025
0
−100
0.06
0.03
NZD/GBP 60 mins
0
0.1
0.05
−10
0.2
0.1
0
−20
0.3
0.15
0
−20
0.4
GBP/EUR 5 mins
0.2
0
−50
0.02
0.01
50
0.04
0.02
0
0.06
0.03
−50
0.08
0.04
0
−100
0.1
GBP/EUR 60 mins
0.05
Figure A.18: Pre-Fix Rate Change Densities
0
−20
0.05
0.1
0.15
0.2
0.25
0
−50
0.02
0.04
0.06
0.08
0
−20
0.05
0.1
0.15
0.2
0.25
0
−50
0.02
0.04
0.06
0.08
−10
−10
10
0
AUD/USD 1 min
0
10
AUD/USD 15 mins
0
JPY/GBP 1 min
0
JPY/GBP 15 mins
20
50
20
50
22
10
20
−10
−50
0
NOK/USD 5 mins
0
10
50
NOK/USD 60 mins
20
100
0
−20
0.05
0.1
0.15
0.2
0
−50
0.01
0.02
0.03
0.04
0.05
0.06
0
10
−10
0
NOK/USD 1 min
0
10
NOK/USD 15 mins
−10
CAD/USD 1 min
0
CAD/USD 15 mins
20
50
20
50
0
−20
0.02
0.04
0.06
0.08
0
−100
0.005
0.01
0.015
0.02
0.025
0
−20
0.02
0.04
0.06
0.08
0.1
0.12
0
−100
0.01
0.02
0.03
0.04
−10
−50
−10
−50
0
SEK/USD 5 mins
0
SEK/USD 60 mins
0
DKK/USD 5 mins
0
DKK/USD 60 mins
10
50
10
50
20
100
20
100
Notes: Densities of price changes (in basis points) away from Fix (black) intra-month pre-Fix (blue) and end-of-month pre-Fix (red).
0
−20
0.02
0.04
0.06
0.08
0.1
0
−100
0.005
0.01
0.015
0.02
0.025
0
−20
0.1
0.05
0
−20
0.2
0
−50
0.1
0
100
0.3
−10
50
0.15
CAD/USD 5 mins
0
0.02
0.04
0.06
0.08
0.1
0.4
−50
CAD/USD 60 mins
0.2
0
−100
0.01
0.02
0.03
0.04
Figure A.19: Pre-Fix Rate Change Densities
0
−20
0.05
0.1
0.15
0.2
0
−50
0.01
0.02
0.03
0.04
0.05
0
−20
0.1
0.2
0.3
0.4
0
−50
0.02
0.04
0.06
0.08
−10
−10
0
SEK/USD 1 min
0
SEK/USD 15 mins
0
DKK/USD 1 min
0
DKK/USD 15 mins
10
10
20
50
20
50
23
100
20
−10
0
SGD/USD 1 min
0
10
SGD/USD 15 mins
20
50
Notes: Densities of price changes (in basis points) away from Fix (black) intra-month pre-Fix (blue) and end-of-month pre-Fix (red).
0
−20
0.1
0.05
10
0.2
0.1
0
0.3
0.15
−10
0.4
0.2
0
−20
0.5
0
−50
0.25
SGD/USD 5 mins
50
0.05
0.02
0
0.1
0.04
−50
0.15
0.06
0
−100
0.2
SGD/USD 60 mins
0.08
Figure A.20: Pre-Fix Rate Change Densities
post
post
post
20
40
0
5
10
−20
−20
20
40
0
5
10
−20
−20
−10
−10
10
20
−20
−20
0
pre
NOK/EUR 5 mins
0
pre
−10
0
10
20
−5
−10
−20
NOK/EUR 15 mins
−10
0
10
20
−40
−40
−20
0
20
40
−10
−10
10
20
−20
−20
0
pre
CHF/EUR 5 mins
0
pre
−10
0
10
20
−5
−10
−20
CHF/EUR 15 mins
−10
0
10
20
−40
−40
−20
0
20
40
−5
0
pre
CHF/EUR 1 mins
0
pre
0
pre
NOK/EUR 1 mins
0
pre
5
10
5
10
NOK/EUR 10 mins
−10
−5
−10
CHF/EUR 10 mins
10
20
10
20
−20
−20
−10
0
10
20
−40
−40
−20
0
20
40
−20
−20
−10
0
10
20
−40
−40
−20
0
20
40
0
pre
JPY/EUR 5 mins
0
pre
10
20
−10
−20
0
pre
NOK/EUR 5 mins
0
pre
10
20
NOK/EUR 15 mins
−10
−20
JPY/EUR 15 mins
Figure A.21: Bivariate Pre- and Post- Fix Rate Change Density
20
40
20
40
−10
−10
−5
0
5
10
−20
−20
−10
0
10
20
−10
−10
−5
0
5
10
−20
−20
−10
0
10
20
−5
0
pre
JPY/EUR 1 mins
0
pre
0
pre
NOK/EUR 1 mins
0
pre
5
10
5
10
NOK/EUR 10 mins
−10
−5
−10
JPY/EUR 10 mins
10
20
10
20
Notes: Each plot shows the contours of the estimated bivariate density for pre- and post-fix price changes (in basis points) over horizons of 1 to 15 minutes. The solid line in each
plot is the estimated projection of the post-fix price change in the pre-fix change. All estimates are based on end-of-month data.
post
post
post
post
post
post
post
post
post
post
post
post
post
24
post
post
post
20
40
0
5
10
−20
−20
20
40
0
5
10
−20
−20
−10
−10
10
20
−20
−20
0
pre
AUD/GBP 5 mins
0
pre
−10
0
10
20
−5
−10
−20
AUD/GBP 15 mins
−10
0
10
20
−40
−40
−20
0
20
40
−10
−10
10
20
−20
−20
0
pre
SEK/EUR 5 mins
0
pre
−10
0
10
20
−5
−10
−20
SEK/EUR 15 mins
−10
0
10
20
−40
−40
−20
0
20
40
−5
−10
−5
−10
0
pre
AUD/GBP 1 mins
0
pre
5
10
5
10
AUD/GBP 10 mins
0
pre
SEK/EUR 1 mins
0
pre
SEK/EUR 10 mins
10
20
10
20
−20
−20
−10
0
10
20
−40
−40
−20
0
20
40
−20
−20
−10
0
10
20
−40
−40
−20
0
20
40
−10
−20
−10
−20
0
pre
CHF/GBP 5 mins
0
pre
CHF/GBP 15 mins
0
pre
AUD/GBP 5 mins
0
pre
10
20
10
20
AUD/GBP 15 mins
Figure A.22: Bivariate Pre- and Post- Fix Rate Change Density
20
40
20
40
−10
−10
−5
0
5
10
−20
−20
−10
0
10
20
−10
−10
−5
0
5
10
−20
−20
−10
0
10
20
−5
−10
−5
−10
0
pre
CHF/GBP 1 mins
0
pre
CHF/GBP 10 mins
0
pre
AUD/GBP 1 mins
0
pre
5
10
5
10
AUD/GBP 10 mins
10
20
10
20
Notes: Each plot shows the contours of the estimated bivariate density for pre- and post-fix rate changes (in basis points) over horizons of 1 to 15 minutes. The solid line in each
plot is the estimated projection of the post-fix rate change in the pre-fix change. All estimates are based on end-of-month data.
post
post
post
post
post
post
post
post
post
post
post
post
post
25
post
post
post
20
40
0
5
10
−20
−20
20
40
0
5
10
−20
−20
−10
−10
10
20
−20
−20
0
pre
NZD/GBP 5 mins
0
pre
−10
0
10
20
−5
−10
−20
NZD/GBP 15 mins
−10
0
10
20
−40
−40
−20
0
20
40
−10
−10
10
20
−20
−20
0
pre
GBP/EUR 5 mins
0
pre
−10
0
10
20
−5
−10
−20
GBP/EUR 15 mins
−10
0
10
20
−40
−40
−20
0
20
40
−5
−10
−5
−10
0
pre
NZD/GBP 1 mins
0
pre
NZD/GBP 10 mins
0
pre
GBP/EUR 1 mins
0
pre
5
10
5
10
GBP/EUR 10 mins
10
20
10
20
−20
−20
−10
0
10
20
−40
−40
−20
0
20
40
−20
−20
−10
0
10
20
−40
−40
−20
0
20
40
−10
−20
−10
−20
10
20
0
pre
AUD/USD 5 mins
0
pre
10
20
AUD/USD 15 mins
0
pre
JPY/GBP 5 mins
0
pre
JPY/GBP 15 mins
Figure A.23: Bivariate Pre- and Post- Fix Rate Change Density
20
40
20
40
−10
−10
−5
0
5
10
−20
−20
−10
0
10
20
−10
−10
−5
0
5
10
−20
−20
−10
0
10
20
−5
−10
−5
−10
0
pre
AUD/USD 1 mins
0
pre
5
10
5
10
AUD/USD 10 mins
0
pre
JPY/GBP 1 mins
0
pre
JPY/GBP 10 mins
10
20
10
20
Notes: Each plot shows the contours of the estimated bivariate density for pre- and post-fix rate changes (in basis points) over horizons of 1 to 15 minutes. The solid line in each
plot is the estimated projection of the post-fix rate change in the pre-fix change. All estimates are based on end-of-month data.
post
post
post
post
post
post
post
post
post
post
post
post
post
26
post
post
post
20
40
0
5
10
−20
−20
20
40
0
5
10
−20
−20
−10
−10
10
20
−20
−20
0
pre
NOK/USD 5 mins
0
pre
−10
0
10
20
−5
−10
−20
NOK/USD 15 mins
−10
0
10
20
−40
−40
−20
0
20
40
−10
−10
10
20
−20
−20
0
pre
CAD/USD 5 mins
0
pre
−10
0
10
20
−5
−10
−20
CAD/USD 15 mins
−10
0
10
20
−40
−40
−20
0
20
40
−5
0
pre
CAD/USD 1 mins
0
pre
0
pre
NOK/USD 1 mins
0
pre
5
10
5
10
NOK/USD 10 mins
−10
−5
−10
CAD/USD 10 mins
10
20
10
20
−20
−20
−10
0
10
20
−40
−40
−20
0
20
40
−20
−20
−10
0
10
20
−40
−40
−20
0
20
40
−10
−20
−10
−20
0
pre
SEK/USD 5 mins
0
pre
SEK/USD 15 mins
0
pre
DKK/USD 5 mins
0
pre
DKK/USD 15 mins
10
20
10
20
Figure A.24: Bivariate Pre- and Post- Fix Rate Change Density
20
40
20
40
−10
−10
−5
0
5
10
−20
−20
−10
0
10
20
−10
−10
−5
0
5
10
−20
−20
−10
0
10
20
−5
−10
−5
−10
0
pre
SEK/USD 1 mins
0
pre
SEK/USD 10 mins
0
pre
DKK/USD 1 mins
0
pre
DKK/USD 10 mins
5
10
5
10
10
20
10
20
Notes: Each plot shows the contours of the estimated bivariate density for pre- and post-fix rate changes (in basis points) over horizons of 1 to 15 minutes. The solid line in each
plot is the estimated projection of the post-fix rate change in the pre-fix change. All estimates are based on end-of-month data.
post
post
post
post
post
post
post
post
post
post
post
post
post
27
28
post
20
40
10
20
−10
−10
0
pre
−20
−20
0
5
10
−5
−10
SGD/USD 5 mins
−10
0
10
20
−20
−20
0
pre
−40
−40
−20
−10
0
10
20
−20
0
20
SGD/USD 15 mins
−5
−10
0
pre
SGD/USD 1 mins
0
pre
5
10
SGD/USD 10 mins
10
20
Figure A.25: Bivariate Pre- and Post- Fix Rate Change Density
Notes: Each plot shows the contours of the estimated bivariate density for pre- and post-fix rate changes (in basis points) over horizons of 1 to 15 minutes. The solid line in each
plot is the estimated projection of the post-fix rate change in the pre-fix change. All estimates are based on end-of-month data.
post
40
post
post
29
−45
−45
−30
−30
−15
−15
0
JPY/EUR
0
15
15
30
30
45
45
60
60
−60
−30
−20
−10
0
10
20
30
−60
−20
−15
−10
−5
0
5
10
15
20
−45
−45
−30
−30
−15
−15
0
NZD/EUR
0
NOK/EUR
15
15
30
30
45
45
60
60
Notes: Average rate path in basis points around 3:45 pm level conditioned on: (i) positive pre-fix changes (over 15 mins) at end of month (solid black); (ii) negative pre-fix changes
(over 15 mins) at end of month (dashed black); (iii) pre-fix changes above the 75th. percentile of end-of-month distribution (upper red dashed dot); (iv) pre-fix changes in the 25th.
percentile of end-of-month distribution (lower red dashed dot); (v) positive and negative pre-fix changes on intra-month days (upper and lower blue dots).
−30
−60
−20
−10
0
10
20
30
−60
−15
−10
−5
0
5
10
15
CHF/EUR
Figure A.26: Rate Paths Around the Fix
30
−45
−45
−30
−30
−15
−15
0
CAD/GBP
0
SEK/EUR
15
15
30
30
45
45
60
60
−60
−20
−15
−10
−5
0
5
10
15
20
−60
−30
−20
−10
0
10
20
30
−45
−45
−30
−30
−15
−15
0
CHF/GBP
0
AUD/GBP
15
15
30
30
45
45
60
60
Notes: Average rate path in basis points around 3:45 pm level conditioned on: (i) positive pre-fix changes (over 15 mins) at end of month (solid black); (ii) negative pre-fix changes
(over 15 mins) at end of month (dashed black); (iii) pre-fix changes above the 75th. percentile of end-of-month distribution (upper red dashed dot); (iv) pre-fix changes in the 25th.
percentile of end-of-month distribution (lower red dashed dot); (v) positive and negative pre-fix changes on intra-month days (upper and lower blue dots).
−60
−30
−20
−10
0
10
20
30
−25
−60
−20
−15
−10
−5
0
5
10
15
20
25
Figure A.27: Rate Paths Around the Fix
31
−45
−45
−30
−30
−15
−15
0
NZD/GBP
0
15
15
30
30
45
45
60
60
−60
−25
−20
−15
−10
−5
0
5
10
15
20
25
−60
−30
−20
−10
0
10
20
30
−45
−45
−30
−30
−15
−15
0
AUD/USD
0
JPY/GBP
15
15
30
30
45
45
60
60
Notes: Average rate path in basis points around 3:45 pm level conditioned on: (i) positive pre-fix changes (over 15 mins) at end of month (solid black); (ii) negative pre-fix changes
(over 15 mins) at end of month (dashed black); (iii) pre-fix changes above the 75th. percentile of end-of-month distribution (upper red dashed dot); (iv) pre-fix changes in the 25th.
percentile of end-of-month distribution (lower red dashed dot); (v) positive and negative pre-fix changes on intra-month days (upper and lower blue dots).
−60
−30
−20
−10
0
10
20
30
−60
−20
−15
−10
−5
0
5
10
15
20
GBP/EUR
Figure A.28: Rate Paths Around the Fix
32
−45
−45
−30
−30
−15
−15
0
NOK/USD
0
15
15
30
30
45
45
60
60
−40
−60
−30
−20
−10
0
10
20
30
40
−60
−15
−10
−5
0
5
10
15
−45
−45
−30
−30
−15
−15
0
SEK/USD
0
DKK/USD
15
15
30
30
45
45
60
60
Notes: Average rate path in basis points around 3:45 pm level conditioned on: (i) positive pre-fix changes (over 15 mins) at end of month (solid black); (ii) negative pre-fix changes
(over 15 mins) at end of month (dashed black); (iii) pre-fix changes above the 75th. percentile of end-of-month distribution (upper red dashed dot); (iv) pre-fix changes in the 25th.
percentile of end-of-month distribution (lower red dashed dot); (v) positive and negative pre-fix changes on intra-month days (upper and lower blue dots).
−60
−25
−20
−15
−10
−5
0
5
10
15
20
25
−60
−20
−15
−10
−5
0
5
10
15
20
CAD/USD
Figure A.29: Rate Paths Around the Fix
33
−45
−30
−15
0
15
30
45
60
Notes: Average rate path in basis points around 3:45 pm level conditioned on: (i) positive pre-fix changes (over 15 mins) at end of month (solid black); (ii) negative pre-fix changes
(over 15 mins) at end of month (dashed black); (iii) pre-fix changes above the 75th. percentile of end-of-month distribution (upper red dashed dot); (iv) pre-fix changes in the 25th.
percentile of end-of-month distribution (lower red dashed dot); (v) positive and negative pre-fix changes on intra-month days (upper and lower blue dots).
−60
−8
−6
−4
−2
0
2
4
6
8
SGD/USD
Figure A.30: Rate Paths Around the Fix